Category: Envision Math

Go through the enVision Math Common Core Grade 7 Answer Key Topic 6 Use Sampling To Draw Inferences About Populations regularly and improve your accuracy in solving questions.

enVision Math Common Core 7th Grade Answers Key Topic 6 Use Sampling To Draw Inferences About Populations

TOPIC 6 GET READY!

Review What You Know!

Vocabulary

Choose the best term from the box to complete each definition.

Question 1.
A ____ is how data values are arranged.
Answer:
Data Distribution,

Explanation:
A data distribution is how data values are arranged.

Question 2.
The part of a data set where the middle values are concentrated is called the ___ of the data
Answer:
Center,

Explanation:
The part of a data set where the middle values are concentrated is called the center of the data.

Question 3.
A ___ anticipates that there will be different answers when gathering information.
Answer:
Statistical Question,

Explanation:
A statistical question anticipates that there will be different answers when gathering information.

Question 4.
____ is a measure that describes the spread of values in a data set.
Answer:
Variability,

Explanation:
Variability is a measure that describes the spread of values in a data set.

Statistical Measures

Use the following data to determine each statistical measure.
9, 9, 14, 7, 12, 8, 11, 19, 15, 11

Question 5.
mean : Add all the given numbers, then divide by the amount of numbers
Answer:
Mean is 11.5,

Explanation :
Sum or total = 9+9+14+7+12+8+11+19+15+11=115 (total of given numbers)
MEAN = 115/10=11.5.

Question 6.
median :
7,8,9,9,11,11,12,14,15,19
Answer:
Median is Eleven (11),

Explanation:
Put the numbers from smallest to the largest, the number in the middle is Median,
if two numbers are in the middle, then add them and divide by 2.
Median = (11+11)/2 = 22/2 = 11.

Question 7.
range
Answer:
Range is Twelve (12),

Explanation:
The range is the difference between the lowest to the highest value in the
given numbers 7,8,9,9,11,11,12,14,15,19,
Lowest number is : 7
Highest number is : 19
Range = 19 – 7 = 12.

Question 8.
mode
Answer:
Mode is Two (2),

Explanation:
The value around which there is the greatest concentration is called mode.
Count how many of each value appears in the given numbers or values
Here the modes are 2, that is 9, 9 and 11,11
Mode =3(Median) – 2(Mean),
= 3(11) – 2(11.5),
= 33 – 23,
= 10.

Question 9.
inter quartile range (IQR)
Answer:
Inter quartile Range is Five (5),

Explanation:
Given statistical measures : 9, 9, 14, 7, 12, 8, 11, 19, 15, 11
Arranged in order 7,8,9,9,11,11,12,14,15,19
First half 7,8,9,9,11,
second half  11,12,14,15,19.

Quartiles : The observations which divide the whole set of observations into four equal parts.
lower quartile LQ : Mid number of or median of a given series
First half is lower quartile LQ = 9,

Upper quartile UQ : Mid number of or median of a given series
second half is lower quartile, UQ = 14
interquartile range(IQR) : The difference between the upper quartile and
the lower quartile is called the interquartile range(IQR)
IQR= UQ – LQ = 14 – 9 = 5.

Question 10.
mean absolute deviation (MAD)
Answer:
MAD is 2.8,

Explanation:
The mean absolute deviation of a data set is the average distance between
each data point and the mean. It gives us an idea about the variability in a data set

Step 1:
Calculate the mean.
sum or total = 9+9+14+7+12+8+11+19+15+11=115 (total of given numbers)
MEAN = 115/10=11.5,

Step 2:
 Calculate how far away each data point is from the mean using positive distances.
These are called absolute deviations.
Data Point    Distance from Mean
1)  9              I   9 – 11.5   I    = 2.5
2)  9              I   9 – 11.5   I    = 2.5
3)  14            I   14 – 11.5   I    = 2.5
4)  7              I    7 – 11.5   I    = 4.5
5)  12            I   12 – 11.5   I   =0.5
6)  8              I   8 – 11.5   I    = 3.5
7)  11            I 11 – 11.5 I    = 0.5
8)  19           I 19 – 11.5 I    = 7.5
9)  15           I 15 – 11.5 I    = 3.5
10)  11           I 11 – 11.5 I   = 0.5

Total measure is 2.5+2.5+2.5+4.5+0.5+3.5+0.5+7.5+3.5+0.5=28.0,
Divide the total measure by the number of observation = 28.0/10 =2.8,
MAD = 28/10 = 2.8.

Data Representations
Make each data display using the data from Problems 5-7.

Question 11.
Answer:
Displayed data is collected from problems 5-7,
Box Plot :
a simple way of representing statistical data on a plot in which a rectangle is drawn to
represent the second and third quartiles, usually with a vertical line inside to
indicate the median value. The lower and upper quartiles are shown as horizontal lines
either side of the rectangle.

Explanation:
Box plot:

Question 12.
Dot plot
Answer:
Dot Plot:
A dot plot, also known as a strip plot or dot chart, is a simple form of data visualization
that consists of data points plotted as dots on a graph with an x- and y-axis.
These types of charts are used to graphically depict certain data trends or groupings.

Explanation:

Statistical Questions

Question 13.
Which is NOT a statistical question that might be used to gather data from a certain group?
A. In what state were you born?
B. What is the capital of the United States?
C. How many pets do you have?
D. Do you like strawberry yogurt?

Answer :
B is NOT statistical questions
Explanation :
A statistical question anticipates that there will be different answers when gathering information, where as Capital of United States is the same answer, so the question What is the capital of the United States? might be used to gather data from a certain group.

Language Development

Fill in the graphic organizer. Write each definition in your own words. Illustrate or cite supporting examples.

Answer:

PICK A PROJECT

PROJECT 6A

What types of changes would you like to see in your community?
PROJECT: WRITE TO YOUR REPRESENTATIVE
Answer:

PROJECT 6B
How could you combine physical activity and a fundraiser?
PROJECT: ANALYZE AN ACTIVITY
Answer:
Physical Activity Marathon is one of the fundraiser.
Explanation:

PROJECT 6C
If you could study an animal population in depth, which animal would you choose, and why?
PROJECT: SIMULATE A POPULATION STUDY

Answer : Tiger
After a century of decline, overall wild tiger numbers are starting to tick upward. Based on the best available information, tiger populations are stable or increasing in India, Nepal, Bhutan, Russia and China.

PROJECT 6D
If you were to design a piece of art that moved, how would you make it move?
PROJECT: BUILD A MOBILE

Answer:

Explanation:

Mobiles are free-hanging sculptures that can move in the air. These sculptures are not only artistic, but they are also a great demonstration of balanced forces. If you look at a traditional mobile more closely you will usually notice that it is made of various horizontal rods.

Materials:
Heavy construction paper or cardstock (various colors work well)
Hole punch, Pen, Markers, Scissors, Tape, String, Straws, at least 6
Ceiling or doorframe you can hang the mobile from (and a chair or adult to help in hanging it)

Preparation:
Carefully cut out the different shapes with your scissors. If you like, you can decorate each of them.
Punch a hole into the top center of each of the cut-out shapes.Attach a piece of string to each of the shapes by threading it through the punched hole and tying a knot. Try to vary the length of string attached to each shape so that they are not all the same.

Procedure:
Start with one layer of your mobile. Attach a piece of string to the center of one of your straws. Hold the straw by the string so it is hanging freely in the air. Once the straw is balanced tie your first shape to one end of the straw.Tie a second shape to the other end of the straw then hold the straw up in the air again.Balance the straw by moving one of the shapes along the straw. Use a second straw and two more shapes to build another balanced structure.

Lesson 6.1 Populations and Samples

Solve & Discuss It!

The table shows the lunch items sold on one day at the middle school cafeteria. Use the given information to help the cafeteria manager complete his food supply order for next week.

Generalize
what conclusions can you draw from the lunch data?
Answer:
Highest Sold items are Hot Dog and least sold items are Veggie Burger.
Explanation:
In the middle school cafeteria Highest Sold items are Hot Dog and least sold items are Veggie Burger, this information help the cafeteria manager to complete his food supply order for next week.

Focus on math practices
Construct Arguments Why might it be helpful for the cafeteria manager to look at the items ordered on more than one day?

Sales of food items information to help the cafeteria manager complete his food supply order to avoid wastage of excess food items and the loos occur due to the less or non saleable items.

Essential Question
How can you determine a representative sample of a population?
Answer:
A subset of a population that seeks to accurately reflect the characteristics of the larger group.
Explanation:
In case of Morgan and her friends are sub set of the registered voted of Morgan town for construction of new stadium.

Try It!
Miguel thinks the science teachers in his school give more homework than the math teachers. He is researching the number of hours middle school students in his school spend doing math and science homework each night.
Answer:
The population includes all the students in Miguel’s middle school.
A possible sample is some students from each of the grades in the middle school.

Convince Me!
Why is it more efficient to study a sample rather than an entire population?
Answer:
To study the whole population is often very expensive and time consuming because of the number of people involved.
For example every 10th person of the population and reduce the time by 10 and still get a representative results.

Try It!
A produce manager is deciding whether there is customer demand for expanding the organic food section of her store. How could she obtain the information she needs?
Answer:
The manager can interview the customers random sample for customer demand for expanding the organic food section of her store. As a sample out of the population.

Try It!
Ravi is running against two other candidates for student council president. All of the 750 students in Ravi’s school will vote for student council president. How can Ravi generate a representative sample that will help him determine whether he will win the election?
Answer:
Ravi can ask one in every 10 people to reduce the number of people he needs to interview in order to make a sample out of 750 students, as we know that a random sample 75 students, Ravi generate a representative sample that will help him determine whether he will win the election.

Try It!
The table at the right shows the random sample that Jeremy generated from the same population as Morgan’s and Maddy’s samples. Compare Jeremy’s sample to Morgan’s and Maddy’s.

Answer :
Jeremy’s sample also contains 20samples. and the sample shares the value 36 with Morgan’s sample and one value 126 with Maddy’s sample, the distribution of values in Jeremy’s sample is different then that of Morgans and Maddy.

KEY CONCEPT
A population is an entire group of objects-people, animals, plants—from which data can be collected. A sample is a subset of the population. When you ask a statistical question about a population, it is often more efficient to gather data from a sample of the population.
A representative sample of a population has the same characteristics as the population. Generating a random sample is one reliable way to produce a representative sample of a population. You can generate multiple random samples that are different but that are each representative of the population.

Do You Understand?

Question 1.
Essential Question
How can you determine a representative sample of a population?
Answer:
A representative sample of a population has the same characteristics as the population but generating a random sample.

Question 2.
Construct Arguments Why does a sample need to be representative of a population?
Answer:
A sample is need for Reliability.
Explanation:
A random sample is one reliable way to produce a representative sample of a population.

Question 3.
Be Precise The quality control manager of a peanut butter manufacturing plant wants to ensure the quality of the peanut butter in the jars coming down the assembly line. Describe a representative sampling method she could use.

Answer:
The quality control manager of a peanut butter manufacturing plant must adopt repeat in line quality checking method to ensure of the quality of the peanut butter in the jar coming down the assembly line. the manager or representative must check every 4th peanut butter jars coming down.

Do You Know How?

Question 4.
A health club manager wants to determine whether the members would prefer a new sauna or a new steam room. The club surveys 50 of its 600 members. What is the population of this study?
Answer:
The population of this study is 600.

Question 5.
A journalism teacher wants to determine whether his students would prefer to attend a national writing convention or tour of a local newspaper press. The journalism teacher has a total of 120 students in 4 different classes. What would be a representative sample in this situation?
Answer:
Representative sample is Four(4)
Explanation:
The journalism teacher will prefer 4 representatives, one sample from each class. total 4 representative sample for collecting the student prefer for attending National writing convention or Local newspaper press.

Question 6.
Garret wants to find out which restaurant people think serves the best beef brisket in town.
a. What is the population from which Garret should find a sample?
b. What might be a sample that is not representative of the population?
Answer:
a. A population is an entire group of people in the town visits restaurant for the best beef brisket.

b. A representative sample of a population in a restaurant is who does not prefer the beef brisket.

Practice & Problem Solving

Leveled Practice In 7 and 8, complete each statement with the correct number.

Question 7.
Of a group of 200 workers, 15 are chosen to participate in a survey about the number of miles they drive to work each week.
Answer:
Sample consists of 15 workers out of 500.
Explanation:
In this situation, the sample consists of the 15 workers selected to participate in the survey are random samples.
The population consists of 200 workers.

Question 8.
The ticket manager for a minor league baseball team awarded prizes by drawing four numbers corresponding to the ticket stub numbers of four fans in attendance.
Answer:
In this situation, the sample consists of the 4 people selected to win a prize.
Explanation:
The population consists of 4 the spectators who purchased tickets to attend the game.

Question 9.
A supermarket conducts a survey to find the approximate number of its customers who like apple juice. What is the population of the survey?
Answer:
Representative Sample survey.
Explanation:
A representative sample of a population has the same characteristics as the population. All the population of from that town.

Question 10.
A national appliance store chain is reviewing the performances of its 400 sales associate trainees. How can the store choose a representative sample of the trainees?
Answer:
Random Sample.
Explanation:
 Random sampling is a part of the sampling in which each sample has an equal probability of being chosen. 

Of the 652 passengers on a cruise ship, 30 attended the magic show on board.
a. What is the sample?
b. What is the population?
Answer:
a. sample is 30
b. population is 652
Explanation:
Total number of passengers is equal to the population in ship i.e.,652
Out of which 30 attended magic show, so the sample became 30.

Question 12.
Make Sense and Persevere
The owner of a landscaping company is investigating whether his 120 employees would prefer a water cooler or bottled water. Determine the population and a representative sample for this situation.

Answer:
The Population is 120 employees.
Explanation:
A representative sample is a subset of a population that seeks to accurately reflect the characteristics of the larger group that is 120employees, 12 employees can be considered as representative sample for investigating whether his 120 employees would prefer a water cooler or bottled water.

Question 13.
Higher Order Thinking
A bag contains 6 yellow marbles and 18 red marbles. If a representative sample contains 2 yellow marbles, then how many red marbles would you expect it to contain? Explain.

Answer:
6 marbles are expected.
Explanation:

Question 14.
Chung wants to determine the favorite hobbies among the teachers at his school. How could he generate a representative sample? Why would it be helpful to generate multiple samples?
Answer:
A representative sample is a subset of a population that seeks to accurately reflect the characteristics of the larger group.
It would be helpful to determine the favorite hobbies among teachers.
Explanation:
For example,
A classroom of 30 teachers with 15 males and 15 females could generate a representative sample,
it can help to generate multiple samples, to determine favorite hobbies.

Question 15.
The table shows the results of a survey conducted to choose a new mascot. Yolanda said that the sample consists of all 237 students at Tichenor Middle School.
a. What was Yolanda’s error?
Answer:
Yolanda’s error is the total population is all students must be a round figure 240
Explanation:
Errors happen when you take a sample from the population rather than using the entire population. In other words, it’s the difference between the statistic you measure and the parameter you would find if you took a census of the entire population.
Then 24 out of 240 be a 10% sample
Instead of 24 of 237 is 10.12%
Main reason for sample size in the population is important.

b. What is the sample size? Explain.
Answer:
sample size is 40.
A sample size is a part of the population chosen for a survey or experiment.
Explanation:
For example, you might take a survey of car owner’s brand preferences. You won’t want to survey all the millions of Cars owners in the country, so you take a sample size. That may be several thousand owners. The sample size is a representation of all car owner’s brand preferences. If you choose your sample wisely, it will be a good representation.

Question 16.
Reasoning
To predict the outcome of the vote for the town budget, the town manager assigned random numbers and selected 125 registered voters. He then called these voters and asked how they planned to vote. Is the town manager’s sample representative of the population? Explain.
Answer:
YES, the town manager’s sample representative of the population.
Explanation:
A representative sample is where your sample matches some characteristic of your population, usually the characteristic you’re targeting with your research. the town manager selected 125 registered voters randomly to ask how they plan to vote.

Question 17.
David wants to determine the number of students in his school who like Brussels sprouts. What is the population of David’s study?
Answer:
The population of David’s Study is the the number of students in his school those who like brussels sprouts and dose not like.
Explanation:
A population is a whole, it’s every member of a group. A population is the opposite to a sample, which is a fraction or percentage of a group.

Question 18.
Researchers want to determine the percentage of Americans who have visited The Florida Everglades National Park in Florida. The diagram shows the population of this study, as well as the sample used by the researchers. After their study, the researchers conclude that nearly 75% of Americans have visited the park.
a. What error was likely made by the researchers?

Answer:
A sampling error is a statistical error that occurs when an analyst does not select a sample that represents the entire population of data. As a result, the results found in the sample do not represent the results that would be obtained from the entire population. Here the researchers conclude that nearly 75% of Americans have visited the park.

b. Give an example of steps researchers might take to improve their study.
Answer:
Sampling errors are easy to identify. Here are a few simple steps to reduce sampling error:

1. Increase sample size: A larger sample size results in a more accurate result because the study gets closer to the actual population size.
2. Divide the population into groups: Test groups according to their size in the population instead of a random sample. For example, if people of a specific demographic make up 20% of the population, make sure that your study is made up of this variable.
3. Know your population: Study your population and understand its demographic mix. Know what demographics use your product and service and ensure you only target the sample that matters.

Question 19.
An art teacher asks a sample of students if they would be interested in studying art next year. Of the 30 students he surveys, 81% are already enrolled in one of his art classes this year. Only 11% of the school’s students are studying art this year. Did the teacher survey a representative sample of the students in the school? Explain.

Answer:
The teacher surveys 30 students, total population of the school is not given to infer the survey results.

Question 20.
Make Sense and Persevere
A supermarket wants to conduct a survey of its customers to find whether they enjoy oatmeal for breakfast. Describe how the supermarket could generate a representative sample for the survey.
Answer:
The manager of the super market can interview people at random.
for example in the store every 10th customer use the random sample as a representative sample of the population, enjoy oatmeal for breakfast.

Question 21.
Critique Reasoning
Gwen is the manager of a clothing store. To measure customer satisfaction, she asks each shopper who makes big purchases for a rating of his or her overall shopping experience. Explain why Gwen’s sampling method may not generate a representative sample.
Answer: In general customer visit the store with positive attitude and satisfaction. Here the mistake made by Gwen was sampling big purchases customer for a rating of his or her overall shopping experience rather then considering all the customers of a clothing store.

Assessment Practice

Question 22.
Sheila wants to research the colors of houses on a highly populated street. Which of the following methods could Sheila use to generate a representative sample? Select all that apply.
Assign each house a number and use a random number generator to produce a list of houses for the sample.
Choose every house that has at least 3 trees in the front yard.
Choose only the houses of the people you know.
List the house numbers on slips of paper and draw at least 20% of the numbers out of a box.
Choose all of the houses on the street that have shutters.
Answer:
The statements that apply are
Assign each house a number and use a random number generator to produce a list of houses for the sample
List the house numbers on slips of paper and draw at least 20% of the numbers out of a box.

Question 23.
A national survey of middle school students asks how many hours a day they spend doing homework. Which sample best represents the population?
PART A
A. A group of 941 students in eighth grade in
B. A group of 886 students in sixth grade in a certain county
C. A group of 795 students in seventh grade in different states
D. A group of 739 students in different middle school grade levels from various states

Answer: D
D. A group of 739 students in different middle school grade levels from various states
is correct

PART B
Explain the reasoning for your answer in Part A.
Answer:
Option D is the only answer that covers multiple grades in different states of the country. That way we have the most representative sample among those four.

Lesson 6.2 Draw Inferences from Data

Solve & Discuss It!

The students in Ms. Miller’s class cast their votes in the school-wide vote for which color to paint the cafeteria walls. Based on the data, what might you conclude about how the rest of the school will vote?

Make Sense and Persevere
How many students are in Ms. Miller’s class? How many students voted for each color?
Answer:
30 students are in Ms. Miller’s class.
Explanation:
Number of students voted for each color.
Radical Rule RED = 7
Box plot BLUE = 12
Geometric mean GREEN = 4
y Plane YELLOW = 3
odd Number ORANGE = 4

Focus on math practices
Reasoning How can you determine whether a sample is representative of a population?
Answer :
Reliability
Explanation:
A sample is a subset of the population. A representative sample of a population has the same characteristics as the population. Generating a random sample is one reliable way to produce a representative sample of a population.

Essential Question
How can inferences be drawn about a population from data gathered from samples?
Answer:
By using sample statistics.
Explanation:
A samples are referred to as sample statistics while values concerning a population are referred to as population parameters. The process of using sample statistics to make conclusions about population parameters is known as inferential statistics.

Try It!
Dash collects data on the hair lengths of a random sample of seventh-grade boys in his school.
Answer:
The data are clustered between 1/2  and 2 inches and between 3½ and 4½ inches. Dash can infer from the data that seventh-grade boys in his school have both short and long hair.
Convince Me!
How does a dot plot help you make inferences from data?
Answer :
A Dot Plot is a type of simple histogram-like chart used in statistics for relatively small data sets where values fall into a number of discrete bins

Try It!
Alexis surveys three different samples of 20 students selected randomly from the population of 492 students in the seventh grade about their choice for class president. In each sample, Elijah receives the fewest votes. Alexis infers that Elijah will not win the election. Is her inference valid? Answer:
Yes, Her inference valid.
explanation:
More then 10% for population 492 is surveyed by Alexis by selecting 20 students of 3 groups is total 60 students, Elijah receives less votes.

Try It!
For his report, Derek also collects data from a random sample of eighth graders in his school, and finds that 18 out of 20 eighth graders have cell phones. If there are 310 eighth graders in his school, estimate the number of eighth graders who have cell phones.
Answer:
The number of eighth garden who have cell phones are 279.
Explanation:

KEY CONCEPT

You can analyze numerical data from a random sample to draw inferences about the population. Measures of center, like mean and median, and measures of variability, like range, can be used to analyze the data in a sample.

Do You Understand?

Question 1.
Essential Question
How can inferences be drawn about a population from data gathered from samples?
Answer:
Inferential statistics is a way of making inferences about populations based on samples.
Explanation:
The inferences about the population Hours of sleep per night is 9pm to 9:30pm has more population

Question 2.
Reasoning Why can you use a random sample to make an inference?
Answer:
A random sample is the subset of the population selected without bias in order to make inferences about the entire population.
Explanation:
Random samples are more likely to contain data that can be used to make predictions about a whole population. The size of a sample influences the strength of the inference about the population.

Question 3.
Critique Reasoning
Darrin surveyed a random sample of 10 students from his science class about their favorite types of TV shows. Five students like detective shows, 4 like comedy shows, and 1 likes game shows. Darrin concluded that the most popular type of TV show among students in his school is likely detective shows. Explain why Darrin’s inference is not valid.
Answer:
Darrin’s inference is not valid because he concluded on most of the students like detective shows.
Explanation:
Out of 10 sample students
5 like – detective shows
4 like – comedy shows
1 like – game shows
Darrin’s concluded on most of the students like detective shows.

Do You Know How?

Question 5.
In a carnival game, players get 5 chances to throw a basketball through a hoop. The dot plot shows the number of baskets made by 20 different players.

a. Make an inference by looking at the shape of the data.
Answer:
Total 20 players, each player get 5 chance, except 2 players, 18 players throw a basketball through a hoop successfully.
Explanation:
2 players – Zero out of five score
3 players – 5 out of five score
3 players – 1 out of five score
4 players – 2 out of five score
4 players – 3 out of five score
4 players – 4 out of five score

 

b. What is the median of the data? What is the mean? Do these measures of center support the inference you made in part (a)?
Answer:
Median =  10
Explanation:
Throw a basketball through a hoop if we arrange in ascending order 0,3,8,12,15,16 as the measures, and the average of 8 and 12 will be the median
(8+12)/2=20/2=10

Answer:
Mean = 9
Explanation:
If we add all the measures and divided by the number as shown below
(0+3+8+12+15+16)/6=54/6=9

Question 6.
In the dot plot above, 3 of 20 players made all 5 baskets. Based on this data, about how many players out of 300 players will make all 5 baskets?
Answer:
45 players will make all 5 baskets.
Explanation:
3 of 20 players made 5 baskets
X of 300 players will make 5 baskets
cross multiply as shown below

Question 7.
The manager of a box office gathered data from two different ticket windows where tickets to a music concert were being sold. Does the data shown in the box plots below support the inference that most of the tickets sold were about $40? Explain.

Answer:
NO, the box plot will not support the inference.
Explanation :
As per the Box Plot most of the tickets sold were about $50 to $60 as IQR or Q2 lies between the 50-60

Practice & Problem Solving

Leveled Practice In 8-10, use the sample data to answer the questions.

Alicia and Thea are in charge of determining the number of T-shirts to order to sell in the school store. Each student collected sample data from the population of 300 students. Alicia surveyed 50 students in the cafeteria. Thea surveyed the first 60 students who arrived at school one morning.

Question 8.
Use Alicia’s data to estimate the number of T-shirts they should order.

Answer:
180 T- shirts should be order.
Explanation:

They should order about 180 T-shirts.

Question 9.
Use Thea’s data to estimate the number of T-shirts they should order.

Answer:
255 T-shirts should be order.
Explanation:

They should order about 255 T-shirts

Question 10.
Construct Arguments Can Alicia or Thea make a valid inference? Explain.
Answer:
Thea : As per my survey 255 students like T- shirts of 300 students
Alicia: As per my survey 180 T-shirts to be ordered for sale of 300 students

Question 11.
Three of the five medical doctors surveyed by a biochemist prefer his newly approved Brand X as compared to the leading medicine. The biochemist used these results to write the TV advertisement shown. Is the inference valid? Explain your answer.

Answer:
Yes, it is valid
Explanation:
60% biochemist survey results approved Brand X as compared to the leading medicine.

Question 12.
Aaron conducted a survey of the type of shoes worn by a random sample of students in his school. The results of his survey are shown at the right.

a. Make a valid inference that compares the number of students who are likely to wear gym shoes and those likely to wear boots.
b. Make a valid inference that compares the number of students who are likely to wear boots and those likely to wear sandals.
Answer:
a) number of students who are likely to wear gym shoes three times more then those likely to wear boots.
b) number of students who are likely to wear boots are two times less then those likely to wear sandals.

Question 13.
Shantel and Syrus are researching the types of novels that people read. Shantel asks every ninth person at the entrance of a mall. She infers that about 26% of the population prefers fantasy novels. Syrus asks every person in only one store. He infers that about 47% of the population prefers fantasy novels.
a. Construct Arguments Whose inference is more likely to be valid? Explain.
b. What mistake might Syrus have made?
Answer:
a) Shantel asks every ninth person at the entrance of a mall is the correct sample survey for researching the type of novels the people read, which gives 26% of the population prefers fantasy novels.
b) Syrus askes in only one store can not give good results even he infers 47% of the population prefers the fantasy navels.

Question 14.
Higher Order Thinking A national TV news show conducted an online poll to find the nation’s favorite comedian. The website showed the pictures of 5 comedians and asked visitors of the site to vote. The news show inferred that the comedian with the most votes was the funniest comedian in the nation.

a. Is the inference valid? Explain.
b. How could you improve the poll? Explain.
Answer:
YES, its valid.
Explanation : conducting survey by a national TV news show by online poll to find the nation’s favorite comedian. the comedian #3 with the most votes was the funniest comedian in the nation.
Broadcasting the news in other channels can participate more population for more accurate results.
In 15 and 16, use the table of survey results from a random sample of people about the way they prefer to view movies.

Question 15.
Lindsay infers that out of 400 people, 300 would prefer to watch movies in a theater. Is her inference valid? Explain.

Answer:
NO, her inference is not valid.
Explanation :
As per survey results form a random sample of people preference is given to Streaming rather then Theater.

Question 16.
Which inferences are valid? Select all that apply.
Going to a theater is the most popular way to watch a movie.
About twice as many people would prefer to stream movies instead of watching in a theater.
About 3 times as many people would prefer to watch a movie on DVD instead of watching in a theater.
About 8 times as many people would prefer to watch a movie on DVD instead of streaming.
Most people would prefer streaming over any other method.
Answer:
Most people would prefer streaming over any other method

Question 17.
Monique collects data from a random sample of seventh graders in her school and finds that 10 out of 25 seventh graders participate in after-school activities. Write and solve a proportion to estimate the number of seventh graders, n, who participate in after-school activities if 190 seventh graders attend Monique’s school.
Answer:
76 students participated.
Explanation:

Question 18.
Each of the 65 participants at a basketball camp attempted 20 free throws. Mitchell collected data for the first 10 participants, most of whom were first-time campers. Lydia collected data for the next 10 participants, most of whom had attended the camp for at least one week.

a. Using only his own data, what inference might Mitchell make about the median number of free throws made by the 65 participants?
Answer:
Median is 9 as per the Mitchell data
Explanation:
IQR remains same even more entries raise.
b. Using only her own data, what inference might Lydia make about the median number of free throws made by the 65 participants?
Answer:
Median is 12 as per the Lydia’s Data
Explanation IQR remain same

c. Who made a valid inference? Explain.
Answer:
Both Mitchell and Lydia made a valid inference.
Explanation:
Mitchell collected data of first 10 participants, most of whom were first-time campers. Lydia collected data for the next 10 participants, most of whom had attended the camp for at least one week.

Assessment Practice

Question 19.
June wants to know how many times most people have their hair cut each year. She asks two of her friends from Redville and Greenburg, respectively, to conduct a random survey. The results of the surveys are shown below.
Redville surveyed on 50 people
Median number of haircuts: 7
Mean number of haircuts: 7.3

Greenburg: 60 people surveyed
Median number of haircuts: 6.5
Mean number of haircuts: 7.6
June infers that most people get 7 haircuts per year. Based on the survey results, is this a valid inference? Explain.
Answer:
YES, its valid inference.
Explanation :
The mean and median are the averages of the survey measures or data collected and number 7 lies in between the 6.5 to 7.6 its a valid

TOPIC 6 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary Krista says that her chickens lay the most eggs of any chickens in the county. To prove her claim, she could survey chicken farms to see how many eggs each of their chickens laid that day. In this scenario, what is the population and what is a possible representative sample?
Answer:
A population is an entire group of objects-people, animals, plants—from which data can be collected.
A representative sample of a population has the same characteristics as the population.

Question 2.
Marcy wants to know which type of book is most commonly checked out by visitors of her local public library. She surveys people in the children’s reading room between 1:00 and 2:00 on Saturday afternoon. Select all the statements about Marcy’s survey that are true. Lesson 6-1
Marcy’s sample is not representative because not all of the library’s visitors go to the children’s reading room.
Marcy’s sample is a representative sample of the population.
Marcy will get a random sample by surveying as many people in the children’s reading room as possible.
The population of Marcy’s study consists of all visitors of the public library.
The results of Marcy’s survey include a mode, but neither a mean nor a median.

For Problems 3-5, use the data from the table.

Question 3.
Michael surveyed a random sample of students in his school about the number of sports they play. There are 300 students in Michael’s school. Use the results of the survey to estimate the number of students in Michael’s school who play exactly one sport. Explain your answer. Lesson 6-2

Answer:
45 students play exactly one sport
Explanation:

 

Question 4.
What inference can you draw about the number of students who play more than one sport? Lesson 6-2
Answer:
160 students play more then 1 sport.
Explanation:

Question 5.
Avi says that Michael’s sample was not random because he did not survey students from other schools. Is Avi’s statement correct?
Explain. Lesson 6-1
Answer:
No, Avi’s statement is not correct.
Explanation:
Michael’s survey is about his school students and their play, not about other school.
How well did you do on the mid-topic checkpoint? Fill in the stars.

Topic 6 MID-TOPIC PERFORMANCE TASK

Sunil is the ticket manager at a local soccer field. He wants to conduct a survey to determine how many games most spectators attend during the soccer season.

PART A
What is the population for Sunil’s survey? Give an example of a way that Sunil could collect a representative sample of this population.
Answer:
The population in Sunil’s survey is 150.
He could collect the sample from ticket manager at a local Soccer field.
Explanation:
According to part B population of sunil’s survey is determined.

PART B
Sunil conducts the survey and obtains the results shown in the table below. What can Sunil infer from the results of the survey?

Answer:
Only 1 or 2 games be conducted.
Explanation:
From the above survey he concluded.

PART C

Suppose 2,400 spectators attend at least one game this soccer season. Use the survey data to estimate the number of spectators who attended 5 or more games this season. Explain how you made your estimate.
Answer:
The survey data to estimate the number of spectators who attended 5 or more games this season is 2,400.

Lesson 6.3 Make Comparative Inferences About Populations

Explore It!

Ella surveys a random sample of 20 seventh graders about the number of siblings they have.

The table shows the results of her survey.

A. Model with Math Draw a model to show how Ella can best display her data.
Answer:

Explanation:
In the above data number of students siblings are displayed on DOT PLOT.
B. Explain why you chose that model.
Dot Plot is easy and effective to show the data.
Explanation:
Dot plot and Box plot are the types of math draw models, that data can be shown in chart format, here Dot plot taken as it is easy and effective way of showing data on charts,

Focus on math practices
Reasoning Using your data display, what can you infer about the number of siblings that most seventh graders have? Explain.
Answer:
Only 1 sibling.
Explanation:
As shown in the above dot plot of 7th grade students has numbers of siblings 1 are more

Essential Question
How can data displays be used to compare populations?
Answer :
Data can be displayed using Dot plot, Box plot or Histogram to compare the population for concluding valid reasons.

Try It!

Kono gathers the heights of a random sample of sixth graders and seventh graders and displays the data in box plots. What can he say about the two data sets?

Answer:
The median of the 7th grade sample is greater than the median of the 6th grade sample.
The 7th grade sample has greater variability.
Explanation:
By comparing both the box plots, he concluded that 7th grade has greater variability. Most 7th grade students have one sibling 8 out of 20 students

Convince Me!
How can you visually compare data from two samples that are displayed in box plots?
Answer:
Guidelines for comparing boxplots from two sam

Try It!

A local recreation center offers a drop-in exercise class in the morning and in the evening. The attendance data for each class over the first month is shown in the box plots at the right. What can you infer about the class attendance?

Answer:
The line for the median of evening attendance data set is to the right of the line for the median of morning attendance data set.
so, morning attendance data can say that the median of evening attendance data set is greater.
Explanation:
The box for evening attendance data set is longer then the morning data set.so, evening attendance data is more spread out or grater variability

KEY CONCEPT
You can use data displays, such as box plots, to make informal comparative inferences about two populations. You can compare the shapes of the data displays or the measures of center and variability.

Do You Understand?

Question 1.
Essential Question How can data displays be used to compare populations?
Answer:
The median and variability of measures are compared between the data set A and set B

Question 2.
Generalize What measures of variability are used when comparing box plots? What do these measures tell you?
Answer:
The median and variability of measures are compared between the data set A and set B.
Explanation:
Here the median is same 5, and the data set B, variability is 9 , that is greater compared to data set A  is 7.
So ,these measures tell us the greater or smaller comparison

Question 3.
Make Sense and Persevere Two data sets both have a median value of 12.5. Data Set A has an interquartile range of 4 and Data Set B has an interquartile range of 2. How do the box plots for the two data sets compare?
Answer:
BOX PLOT
Explanation:

Do You Know How?

The box plots describe the heights of flowers selected randomly from two gardens. Use the box plots to answer 4 and 5.

Question 4.
Find the median of each sample.
Garden Y median = ___ inches
Garden Z median = ___  inches
Answer:
Garden Y median = 6 inches
Garden Z median = 4 inches
Explanation :
In the above box plot the median or IQR is shown in the graph

Question 5.
Make a comparative inference about the flowers in the two gardens.
Answer:
Heights of the flowers in the Garden .
Y is greater then the Garden Z as the median of the Garden Y flowers is right to the Garden Z flowers.
Garden Y is more spread out or grater variability compare to the Garden Z.
Explanation:
Compare the gardens of Y and Z, displayed in the box plot.

Practice & Problem Solving

Leveled Practice For 6-8, complete each statement.

Question 6.
Water boils at lower temperatures as elevation increases. Rob and Ann live in different cities. They both boil the same amount of water in the same size pan and repeat the experiment the same number of times. Each records the water temperature just as the water starts to boil. They use box plots to display their data. Compare the medians of the box plots.
The median of Rob’s data is the median of Ann’s data.
This means Rob is at elevation than Ann.

Explanation:

Question 7.
Liz is analyzing two data sets that compare the amount of food two animals eat each day for one month.

a. The median of Animal 2’s data is than the median of Animal 1’s data
b. Liz can infer that there is variability in the data for Animal 1 than for Animal 2.
c. Liz can infer that Animal generally eats more food.
Answer:

Question 8.
The box plots show the heights of a sample of two types of trees.
The median height of Tree is greater.

Answer:
Tree 1 height is greater then Tree 2.
Explanation:
The median of the Tree 1 is right side of the Tree 2 median,
So the Height of the Tree 1 is greater.

Question 9.
Reasoning A family is comparing home prices in towns where they would like to live. The family learns that the median home price in Hometown is equal to the median home price in Plainfield and concludes that the homes in Hometown and Plainfield are similarly priced.
What is another statistical measure that the family might consider when deciding where to purchase a home?

Answer:
If median is same, then Mean can be a another statistical measure to check for the best option, and variability of the space of the house plot can be compared for the greater one in space.
Explanation:
Mean and median both try to measure the “central tendency” in a data set. The goal of each is to get an idea of a “typical” value in the data set. The mean is commonly used, but sometimes the median is preferred.

Question 10.
Higher Order Thinking The box plots show the daily average high temperatures of two cities from January to December. Which city should you live in if you want a greater variability in temperature? Explain.

Answer:
City X has greater variability.
Explanation :
City X and Y are of same temperature variability, but the median of City X is less then the City Y.

Assessment Practice

Question 11.
Paul compares the high temperatures in City 1 and City 2 for one week. In City 1, the range in temperature is 10°F and the IQR is 5°F. In City 2, the range in temperature is 20°F and the IQR is 5°F. What might you conclude about the weather pattern in each city based on the ranges and interquartile ranges?
A. The weather pattern in City 1 is more consistent than the weather pattern in City 2.
B. The weather patterns in City 1 and City 2 are equally consistent.
C. The weather pattern in City 2 is more consistent than the weather pattern in City 1.
D. The range and interquartile range do not provide enough information to make a conclusion.
Answer:
Option A
Explanation:
The IQR of both the Cities are same (IQR is 5°F) but the rage in temperature city 1 has less then the city 2.

Lesson 6.4 Make More Comparative Inferences About Populations

Explore It!

Jackson and his brother Levi watch Jewel Geyser erupt one afternoon. They record the time intervals between eruptions. The dot plot shows their data.
Jackson estimates that the average time between eruptions is 8 minutes. Levi estimates that the average time between eruptions is 8\(\frac{1}{2}\) minutes.

A. Construct Arguments Construct an argument to support Jackson’s position.
Jackson estimates the average time between eruptions is 8 minutes.
Levi estimates the average time between eruptions is 8\(\frac{1}{2}\) minutes. Jackson estimate is nearer to the median value of the time as shown in the above dot plot

B. Construct Arguments Construct an argument to support Levi’s position.
Jackson estimates the average time between eruptions is 8 minutes. Levi estimates the average time between eruptions is 8\(\frac{1}{2}\) minutes. Levi’s estimate is exactly value of the time  as shown in the above dot plot

Focus on math practices
Reasoning How can you determine the best measure of center to describe a set of data?

Answer:
we can determine the best measure of center to describe a set of data is by finding the Mean and Median of data.
Explanation:
The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.
The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.

 

Essential Question
How can dot plots and statistical measures be used to compare populations?
Answer:
By calculating the Mean and Median of data and variability and range.
Explanation:
The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set.
The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.
 Variability is also referred to as spread, scatter or dispersion.
Range: the difference between the highest and lowest values.

 

Try It!

Quinn also collects data about push-ups. Does it appear that students generally did more push-ups last year or this year? Explain your reasoning.

Answer:
No, it does not give any inference.
Explanation:
The students do less push-ups last year then this year.

Convince Me!
How does the range of these data sets affect the shape of the dot plots?

Answer:
In a dot plot, range is the difference between the values represented by the farthest. left and farthest right dots.
Explanation :
Range is 12 – 3 = 9

Try It!

Peter surveyed a random sample of adults and a random sample of teenagers about the number of hours that they exercise in a typical week. He recorded the data in the table below. What comparative inference can Peter make from the data sets?

Answer:
The mean is 4.4 of adults is less then the Teenagers 7.9 the average the number of hours that they exercise in a typical week is more for teenagers.
Explanation:
Drawing comparative inferences may involve analyzing the data using mean, median, mean absolute deviation, interquartile range, range, and/or mode. In this lesson students will analyze data in different forms and draw informal comparative inferences about the populations involved.

KEY CONCEPT

You can use dot plots to make informal comparative inferences about two populations. You can compare the shapes of the data displays or the measures of center and variability.

The modes of Data Set B are greater than the modes of Data Set A.
The mean of Data Set B is greater than the mean of Data Set A.
You can infer that data points are generally greater in Data Set B.
The ranges and the MADs of the data sets are similar. You can infer that the variabilities of the two data sets are about the same.

Do You Understand?

Question 1.
Essential Question How can dot plots and statistical measures be used to compare populations?

Answer:
The statistical measures of the data of set A and B are compared with reference to the Median, Mode of the sets and the variability of the measures are compared for the population.

Question 2.
Reasoning How can you make predictions using data from samples from two populations?
Answer:
we call this making predictions using random sampling from two population. We basically take data from a random sample of two population and make predictions about the whole population based on that data. Random sample – A sample of a population that is random and the every elements of the population is equally likely to be chosen for the sample.

Question 3.
Construct Arguments Two data sets have the same mean but one set has a much larger MAD than the other. Explain why you may want to use the median to compare the data sets rather than the mean.
Answer:
Both set A and B have the same mean.

Do You Know How?

For 4 and 5, use the information below.

Coach Fiske recorded the number of shots on goal his first-line hockey players made during two weeks of hockey scrimmage.

Question 4.
Find the mean number of shots on goal for each week.
Answer:
Mean of week 1 is 5 and week 2 is 7
Explanation:
Mean  in week 1 is 5
(5+4+6+8+2+3+7)/5=35/7=5

Mean in week 2 is 7
(8+7+9+5+5+7+8)/7=49/7=7

Question 5.
a. Based on the mean for each week, in which week did his first line take more shots on goal?
b. Based on the comparison of the mean and the range for Week 1 and Week 2, what could the coach infer?
Answer:
a. week 2
b. the range in week 1 is more then the week 2 and the mean in week 1 is less then the week 2,
week 2 is better then the week 1 as Coach Fiske recorded the number of shots on goals made during two weeks of hockey.

Practice & Problem Solving

Question 6.
A study is done to compare the fuel efficiency of cars. Cars in Group 1 generally get about 23 miles per gallon. Cars in Group 2 generally get about 44 miles per gallon. Compare the groups by their means. Then make an inference and give a reason the inference might be true.

The mean for Group is less than the mean for Group .
The cars in Group generally are more fuel-efficient.
The cars in Group may be smaller.
Answer:

Question 7.
The dot plot shows a random sample of vertical leap heights of basketball players in two different basketball camps. Compare the mean values of the dot plots. Round to the nearest tenth.

The mean values tell you that participants in Camp jump higher in general.
Answer:
camp 1 mean = 28 in
camp 2 mean = 24 in

Explanation:

Question 8.
A researcher divides some marbles into two data sets. In Data Set 1, the mean mass of the marbles is 13.6 grams. In Data Set 2, the mean mass of the marbles is 14 grams. The MAD of both data sets is 2. What can you infer about the two sets of marbles?
Answer:
The mass of marbles in set 2 is higher then the marbles in set 1 as comparing the mean mass of the marbles of both sets.

Question 9.
Generalize Brianna asks 8 classmates how many pencils and erasers they carry in their bags. The mean number of pencils is 11. The mean number of erasers is 4. The MAD of both data sets is 2. What inference could Brianna make using this data?
Answer:
Total 88 pencils and 32 erasers they carry in their bags.
Explanation:
Pencils =(P)/8 = 11
P= 11×8=88
Erasers = (E)/8=4
E=8×4 = 32

Question 10.
Higher Order Thinking Two machines in a factory are supposed to work at the same speed to pass inspection. The number of items built by each machine on five different days is recorded in the table. The inspector believes that the machines should not pass inspection because the mean speed of Machine X is much faster than the mean speed of Machine Y.

a. Which measures of center and variability should be used to compare the performances of each machine? Explain.
b. Is the inspector correct? Explain.
Answer:
a. Median and IQR are used.
Explanation:

b. YES, the inspector is correct.
Explanation:
The mean speed of Machine X is faster then the Machine Y, they should run same speed but they are varying with an average speed of 2.2.

Assessment Practice

Question 11.
The dot plots show the weights of a random sample of fish from two lakes. Which comparative inference about the fish in the two lakes is most likely correct?

A. There is about the same variation in weight between small and large fish in both lakes.
Answer:
No, there is variation.
Explanation:
Variation of weights in Round Lake is from 15 to 21 ounces, difference is 6 ounce. In South lake weights vary from 11 to 21 ounces, difference is 10 ounce
B. There is less variation in weight between small and large fish in South Lake than between small and large fish in Round Lake.
Answer:
No, there is variation.
Explanation:
Variation in south lake is higher then the Round lake, they differ with 10 ounce and 6 ounce respectively.

C. There is less variation in weight between small and large fish in Round Lake than between small and large fish in South Lake.
Answer:
YES, 6 ounce in Round lake.
Explanation:
Variation in south lake is higher then the Round lake, they differ with 10 ounce and 6 ounce respectively
D. There is greater variability in the weights of fish in Round Lake.
Answer:
No, there is variation.
Explanation:
Less variability in the weights of the fish in the round lake.

3-Act Mathematical Modeling: Raising Money

ACT 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Make a prediction to answer this Main Question.
Answer:

Question 4.
Construct Arguments Explain how you arrived at your prediction.

Answer:

ACT 2

Question 5.
What information in this situation would be helpful to know? How would you use that information?

Answer:

Question 6.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 7.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 8.
What is your answer to the Main Question? Does it differ from your prediction? Explain.

Answer:

ACT 3

Question 9.
Write the answer you saw in the video.
Answer:

Question 10.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?

Answer:

Question 11.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

ACT 3

Extension

Reflect

Question 12.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Answer:

Question 13.
Critique Reasoning Explain why you agree or disagree with each of the arguments in Act 2.

SEQUEL

Question 14.
Use Appropriate Tools You and your friends are starting a new school club. Design a sampling method that is easy to use to help you estimate how many people will join your club. What tools will you use?

Answer:

TOPIC 6 REVIEW

Topic Essential Question

How can sampling be used to draw inferences about one or more populations?

Vocabulary Review

Complete each definition, and then provide an example of each vocabulary word used.

1. A population is a entire group of objects from which data can be collected.

2. Making a conclusion by interpreting data is called making an inference

3. A valid inference is one that is true about a population based on a representative sample.

4 A(n) representative sample accurately reflects the characteristics of an entire population.

Use Vocabulary in Writing

Do adults or teenagers brush their teeth more? Nelson surveys two groups: 50 seventh grade students from his school and 50 students at a nearby college of dentistry. Use vocabulary words to explain whether Nelson can draw valid conclusions.

Concepts and Skills Review

LESSON 6-1 Populations and Samples

Quick Review
A population is an entire group of people, items, or events. Most populations must be reduced to a smaller group, or sample, before surveying. A representative sample accurately reflects the characteristics of the population. In a random sample, each member of the population has an equal chance of being included.

Practice

Question 1.
Anthony opened a new store and wants to conduct a survey to determine the best store hours. Which is the best representative sample?
A. A group of randomly selected people who come to the store in one week
B. A group of randomly selected people who visit his website on one night
C. Every person he meets at his health club one night
D. The first 20 people who walk into his store one day
Answer:
Option A

Question 2.
Becky wants to know if she should sell cranberry muffins at her bakery. She asks every customer who buys blueberry muffins if they would buy cranberry muffins. Is this a representative sample? Explain.
Answer:
A representative sample accurately reflects the characteristics of the population. Those like blueberry muffins may not like cranberry muffins
up to their choice.

Question 3.
Simon wants to find out which shop has the best frozen fruit drink in town. How could Simon conduct a survey with a sample that is representative of the population?
Answer:
A representative sample accurately reflects the characteristics of the population. that means Simon select some representative samples of his friends as a random sample, to find out shop has the best frozen fruit drink in town.

LESSON 6-2 Draw Inferences from Data

Quick Review
An inference is a conclusion about a population based on data from a sample or samples. Patterns or trends in data from representative samples can be used to make valid inferences. Estimates can be made about the population based on the sample data.

Practice

Question 1.
Refer to the example. Polly surveys two more samples. Do the results from these samples support the inference made from the example?

Answer:
In all three samples Polly collected there is the least number of students that do crafts for a hobby
that means that Polly made the correct inference

LESSONS 6-3 AND 6-4
Make Comparative Inferences About Populations | Make More Comparative Inferences About Populations

Quick Review
Box plots and dot plots are common ways to display data gathered from samples of populations. Using these data displays makes it easier to visually compare sets of data and make inferences. Statistical measures such as mean, median, mode, MAD, interquartile range (IQR), and range can also be used to draw inferences when comparing data from samples of two populations.

Practice

Question 1.
The two data sets show the number of days that team members trained before a 5K race.

a. What inference can you draw by comparing the medians?
Answer :
Team B median is higher then the A median, The range of Team A is higher then the Team B of the number of days that team members trained before a 5K race

b. What inference can you draw by comparing the interquartile ranges?
Answer:
Team B IQR is right side of the Team A IQR  in the above box plot of days that team members trained before a 5K race
IQR of Team A is 20 and IQR of Team B is 24

Question 2.
The dot plots show how long it took students in Mr. Chauncey’s two science classes to finish their science homework last night. Find the means to make an inference about the data.

Answer:
i) mean of first period is 38.75 Minutes
ii) mean of second period is 35 Minutes

Explanation:
Mean of first period=(20×2+25×2+30×4+35×3+45×2+50×3+55×4)/20=775/20=38.75

Mean of Second Period=(15×1+20×2+25×4+30×4+35×2+45×3+50×1+55×2+60×1)/20=700/20=35

Second period home work taken less time then first period science home work

 

TOPIC 6 Fluency Practice

Riddle Rearranging

Find each percent change or percent error. Round to the nearest whole percent as needed. Then arrange the answers in order from least to greatest. The letters will spell out the answer to the riddle below.

V
A young tree is 16 inches tall. One year later, it is 20 inches tall. What is the percent increase in height?

Answer : 25%
Explanation:
The Tree grew for 4 inches, we need to find the % of the starting height that equals

A
A ship weighs 7 tons with no cargo. With cargo, it weighs 10.5 tons. What is the percent increase in the weight?
Answer: 50%
Explanation:
The ship’s weight changed for 3.5 tons.
Divide 3.5 by the weight of the ship with no cargo to calculate the percentage increase in the ships weight.

R
The balance of an account is $500 in April. In May it is $440. What is the percent decrease in the balance?
Answer: 12%
Explanation:
The change in the balance 60 out of 500.
divide the two values to calculate the percent decrease in the balance.

B
Ben thought an assignment would take 20 minutes to complete. It took 35 minutes. What is the percent error in his estimate of the time?
Answer : 42.86%
Explanation:
Divide the absolute value of the error (15) by the actual time it took to him to complete the assignment (35).

N
Natalie has $250 in savings. At the end of 6 months, she has $450 in savings. What is the percent increase in the amount of her savings?
Answer: 80%
Explanation:
The absolute increase of the money in Natalie’s bank account is $200 and we know that she started the period with $250 in her account.
Divide the two values to calculate the percent increase of the money in her account.

I
The water level of a lake is 22 feet. It falls to 18 feet during one month. What is the percent decrease in the water level?
Answer: 18%
Explanation :
The water lowered for 4 feet in the fall .
Divide that by the water level before the decrease the to calculate the percent decrease of the water level.

R
Shamar has 215 photos on his cell phone. He deletes some so that only 129 photos remain. What is the percent decrease in the number of photos?

Answer : 40%
Explanation:
Divide the number of pictures Shamar deleted(215-129=86) by the number of pictures Shamar had on his phone before he started deleting them(215).

K
Lita estimates she will read 24 books during the summer. She actually reads 9 books. What is the percent error of her estimate?

Answer: 167%
Explanation:
Divide the value of the absolute error.
Lita made(24-9=15) by the actual number of books she red (9), to calculate the percent error she made.

E
Camden estimates his backpack weighs 9 pounds with his books. It actually weighs 12 pounds. What is the percent error of his estimate?

Answer: 25%
Explanation:
The absolute error Camden made is 3
Divide that by the actual weight of the backpack(12) to calculate the percent error Camden made.

Answer: RIVER BANK

Explanation :
Arrange the values from smallest to largest
12<18<25=25<40<42.86<50<80<167
The letters put together give the solution to the Riddle.

Go through the enVision Math Common Core Grade 8 Answer Key Topic 4 Investigate Bivariate Data regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 4 Investigate Bivariate Data

3-ACT MATH

Reach Out

Reach for the skies! Who in your class can reach the highest? That height depends on how tall each person is and the lengths of their arms.
Now stick your arms out to your sides. Sometimes this horizontal distance is called your wingspan. The wandering albatross can have a wingspan of up to 12 feet. How does your wingspan compare? Think about this during the 3-Act Mathematical Modeling lesson.

enVision STEM Project

Did You Know?
A fishery biologist collects data on fish, such as the size and health of the fish population in a particular body of water.

Largemouth bass and smallmouth bass are the most popular game fish in North America.

Biologists often use tagging studies to estimate fish population, as well as to estimate catch and harvest rates.

The average lifespan of bass is about 16 years, but some have lived more than 20 years.

Research suggests that bass can see red better than any other color on the spectrum.

Your Task: How Many Fish?

Suppose a fishery biologist takes 500 basses from a lake, tags them, and then releases them back into the water. Several days later, the biologist nets a sample of 200 basses, of which 30 are tagged. How many basses are in the lake? You and your classmates will explore how the biologist can use sampling to describe patterns and to make generalizations about the entire population.

Answer:
It is given that
A fishery biologist takes 500 basses from a lake, tags them, and then releases them back into the water. Several days later, the biologist nets a sample of 200 basses, of which 30 are tagged.
So,
The total number of basses = 500 + 200
= 700
The number of basses tagged = 30
So,
After netting, the number of basses = 700 – 30
= 670
Hence, from the above,
We can conclude that after the biologist nets 200 basses,
The total number of basses present are: 670

Topic 4 GET READY!

Review What You Know!

Vocabulary

Choose the best term from the box to complete each definition.

Question 1.
____ is the change in y divided by the change in x.
Answer:
We know that,
“Slope” is the change in y divided by the change in x
Hence, from the above,
We can conclude that the best term to complete the given definition is: Slope

Question 2.
A relationship where for every x units of one quantity there are y units of another quantity is a ____
Answer:
We know that,
A relationship where for every x units of one quantity, there are y units of another quantity is a “Ratio”
Hence, from the above,
We can conclude that the best term to complete the given definition is: Ratio

Question 3.
The ____ is the horizontal line in a coordinate plane.
Answer:
We know that,
The “X-axis” is the horizontal line in a coordinate plane
Hence, from the above,
We can conclude that the best term to complete the given definition is: X-axis

Question 4.
The ___ is the vertical line in a coordinate plane.
Answer:
We know that,
The “Y-axis” is the vertical line in a coordinate plane
Hence, from the above,
We can conclude that the best term to complete the given definition is: Y-axis

Graphing Points
Graph and label each point on the coordinate plane.

Question 5.
(-2, 4)
Answer:

Question 6.
(0, 3)
Answer:

Question 7.
(3, -1)
Answer:

Question 8.
(-4, -3)
Answer:

Finding Slope

Find the slope between each pair of points.

Question 9.
(4, 6) and (-2, 8)
Answer:
The given points are: (4, 6), (-2, 8)
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{8 – 6}{-2 – 4}\)
= \(\frac{2}{-6}\)
= –\(\frac{1}{3}\)
Hence, from the above,
We can conclude that the slope between the given points is: –\(\frac{1}{3}\)

Question 10.
(-1, 3) and (5,9)
Answer:
The given points are: (-1, 3), (5, 9)
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{9 – 3}{5 + 1}\)
= \(\frac{6}{6}\)
= 1
Hence, from the above,
We can conclude that the slope between the given points is: 1

Question 11.
(5, -1) and (-3, -7)
Answer:
The given points are: (5, -1), (-3, -7)
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{-7 + 1}{-3 – 5}\)
= \(\frac{-6}{-8}\)
= \(\frac{3}{4}\)
Hence, from the above,
We can conclude that the slope between the given points is: \(\frac{3}{4}\)

Writing Fractions as Percents

Question 12.
Explain how to write \(\frac{36}{60}\) as a percent.
Answer:
The given fraction is: \(\frac{36}{60}\)
We know that,
1 = 100%
So,
\(\frac{36}{60}\)
= \(\frac{36}{60}\) × 100%
= \(\frac{36 × 100%}{60}\)
= 60%
Hence, from the above,
We can conclude that the value of \(\frac{36}{60}\) as a percent is: 60%

Language Development

Complete the graphic organizer. Write the definitions of the terms in your own words. Use words or a sketch to show an example.

Answer:

Topic 4 PICK A PROJECT

PROJECT 4A

What carnival games do you have a good chance of winning, and why?
PROJECT: BUILD A CARNIVAL GAME

PROJECT 4B
If you had a superpower, what would it be?
PROJECT: SUMMARIZE SUPERHERO DATA

PROJECT 4C
What makes a song’s lyrics catchy?
PROJECT: WRITE A SONG

PROJECT 4D
How does your dream job use math?
PROJECT: RESEARCH A CAREER

Lesson 4.1 Construct and Interpret Scatter Plots

Solve & Discuss It!

Luciana is starting a two-week social media campaign to attract new subscribers to Blaston, a music website for teens. She has the following data from her last campaign to help plan her strategy.

Look for Relationships
How are the number of media posts and the number of subscribers related?
Answer:
The given data is:

If we observe the total data,
We can find that there is no particular pattern for the given data
But, if we observe the given data in parts, then
From 1 – 5 posts per day,
Social media posts per day ∝ New subscribers
From 8 – 10 posts per day,
Social media posts per day ∝ \(\frac{1}{New subscribers}\)
From 6 – 7 posts per day,
There is no pattern between the social media posts per day and the number of subscribers

Based on this data, what should be Luciana’s strategy for the new campaign?
Answer:
Based on the pattern of the given data (As mentioned above),
Luciana’s strategy for the new campaign must be:
The social media posts per day and the new subscribers must be in an increasing trend only

Focus on math practices
Use Structure What patterns do you see in the data from Luciana’s last social media campaign?
Answer:
If we observe the total data,
We can find that there is no particular pattern for the given data
But, if we observe the given data in parts, then
From 1 – 5 posts per day,
Social media posts per day ∝ New subscribers
From 8 – 10 posts per day,
Social media posts per day ∝ \(\frac{1}{New subscribers}\)
From 6 – 7 posts per day,
There is no pattern between the social media posts per day and the number of subscribers

Essential Question
How does a scatter plot show the relationship between paired data?
Answer:
The scatter diagram graphs pairs of numerical data, with one variable on each axis, to look for a relationship between them. If the variables are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.

Try It!
Luciana collects data about the number of entries and the ages of the subscribers who enter the concert giveaway.

The point that represents the data in the fourth column has coordinates
Answer:
The given data is:

Now,
We know that,
The x-axis always represents the independent variables
The y-axis always represents the dependent variables
So,
From the given data,
The independent variable (x) is: Age
The dependent variable (y) is: Number of entries
We know that,
The ordered pair must be in the form of (x, y)
Hence, from the above,
We can conclude that the point that represents the data in the fourth column has coordinates (13, 9)

Convince Me!
Explain how Luciana would choose scales for the x-axis and y-axis.
Answer:
In a graph,
The scales for the x-axis and the y-axis is nothing but the rate of change between the values of x and y
Ex:
The given data is:

From the above data,
The scale for the x-axis is:
Rate of change between the values of x = 11 – 10 (or) 12 – 11 and so on
= 1
The scale for the y-axis is:
Rate of change between the values of y = 9 – 8 (or) 10 – 9
= 1
Hence, from the above,
We can conclude that
The scale for the x-axis is:
1 unit = 1 year
The scale for the y-axis is:
1 unit = 1 Entry

Try It!
Avery also tracks the number of minutes a player plays and the number of points the player scored. Describe the association between the two data sets. Tell what the association suggests.

Answer:
It is given that
Avery also tracks the number of minutes a player plays and the number of points the player scored.
So,
The given scatter plot is:

Now,
From the given scatter plot,
We can observe that the overall pattern is in an increasing trend
But,
When we observe the number of points in the perspective of minutes,
The pattern is in clusters
Now,
At 4 minutes,
The number of points scored is: 0
At 6 and 8 minutes,
The number of points scored is: 2
Between 8 and 14 minutes,
The number of points scored is: 4
Between 10 and 16 minutes,
The number of points scored is: 6

KEY CONCEPT

A scatter plot shows the relationship, or association, between two sets of data.

The y-values increase as the x-values increase.

The y-values decrease as the x-values increase.

There is no consistent pattern between the y-values and the x-values.

Do You Understand?

Question 1.
? Essential Question
How does a scatter plot show the relationship between paired data?
Answer:
The scatter diagram graphs pairs of numerical data, with one variable on each axis, to look for a relationship between them. If the variables are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.

Question 2.
Model with Math
Marcy always sleeps fewer than 9 hours each night and has never scored more than 27 points in a basketball game. A scatter plot suggests that the more sleep she gets, the more she scores. What scales for the axes might be best for constructing the scatter plot?
Answer:
It is given that
Marcy always sleeps fewer than 9 hours each night and has never scored more than 27 points in a basketball game. A scatter plot suggests that the more sleep she gets, the more she scores.
So,
The scales for the axes that might be best is:
For the x-axis:
It is given that Macy always sleeps fewer than 9 hours
So,
The scale might be: 1 unit starting from 9 to 24
For the y-axis:
It is given that Macy never scored more than 27 points in a basketball game
So,
The scale might be: 1 unit starting from 27 to the corresponding last value of x
Hence, from the above,
We can conclude that
For the x-axis,
The scale is:
1 unit = 1 hour
For the y-axis,
The scale is:
1 unit = 1 point

Question 3.
Construct Arguments Kyle says that every scatter plot will have a cluster, gap, and outlier. Is he correct? Explain.
Answer:
We know that,
A scatter plot might have a cluster, a gap, and an outlier or the association of any two but not all three are present
Hence, from the above,
We can conclude that Kyle is not correct

Do You Know How?

Question 4.
Phoebe constructs a scatter plot to show the data. What scales could she use for the x- and y-axes?

Answer:
It is given that
Phoebe constructs a scatter plot to show the data.
Note:
The scatter plot is drawn only for the relations that are functions and we know that the rate of change is constant for a function
Now,
The given data is:

From the given data,
The scale she could use for the x-axis will be 1 unit
The scale she might use for the y-axis will be: 1 unit
Hence, from the above,
We can conclude that
The scale used for the x-axis is:
1 unit = 1 inch (Shoe size)
The scale used for the y-axis is:
1 unit = 1 inch (Height)

Question 5.
Germaine constructs a scatter plot to show how many people visit different theme parks in a month. Why might clusters and outliers be present?

Answer:
It is given that
Germaine constructs a scatter plot to show how many people visit different theme parks in a month
Now,
The given scatter plot is:

From the given scatter plot,
We can observe that the graph is non-linear
We know that,
A cluster is a group of objects, numbers, data points (information), or even people that are located close together
An outlier is a value in a data set that is very different from the other values. That is, outliers are values unusually far from the middle
So,
From the given scatter plot,
We can observe that there are 2 points that are far from the pattern and those points are called “Outliers”
Since the scatter plot is non-linear, the points will be grouped together and the group of points is called “Clusters”

Practice & Problem Solving

Question 6.
Leveled Practice The table shows the racing times in minutes for the first two laps in a race. Complete the scatter plot.

Answer:
It is given that
The table shows the racing times in minutes for the first two laps in a race
Now,
In the graph,
The x-axis represents: The racing times of Lap 1
The y-axis represents: The racing times of Lap 2
Hence,
The completed scatter plot with the x-axis and y-axis is:

Question 7.
The scatter plot represents the prices and number of books sold in a bookstore.
a. Identify the cluster in the scatter plot and explain what it means.

Answer:
It is given that
The scatter plot represents the prices and number of books sold in a bookstore.
Now,
The given scatter plot is:

Now,
We know that,
A cluster is a group of objects, numbers, data points (information), or even people that are located close together
So,
From the above scatter plot,
We can observe that the clusters are present between the intervals of 10 books sold and 20 books sold

b. Generalize How does the scatter plot show the relationship between the data points? Explain.
Answer:
The scatter diagram graphs pairs of numerical data, with one variable on each axis, to look for a relationship between them. If the variables are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line.

Question 8.
The table shows the monthly attendance in thousands at museums in one country over a 12-month period.

a. Complete the scatter plot to represent the data.
Answer:
It is given that
The table shows the monthly attendance in thousands at museums in one country over a 12-month period.
Now,
The given data is:

Hence,
The representation of the scatter plot for the given data is:

b. Identify any outliers in the scatter plot.
Answer:
We know that,
A value that “lies outside” (is much smaller or larger than) most of the other values in a set of data is called an “Outlier”
Hence, from the above,
We can conclude that
The outliers in the scatterplot are at (12, 3), (6, 36)

c. What situation might have caused an outlier?

Answer:
From part (b),
The outliers in the scatterplot are at (12, 3), (6, 36)
So,
From the given points,
We can conclude that the number of people is very low at that particular month to cause the situation of outliers

Question 9.
Higher-Order Thinking The table shows the number of painters and sculptors enrolled in seven art schools. Jashar makes an incorrect scatter plot to represent the data.

a. What error did Jashar likely make?

Answer:
It is given that
The table shows the number of painters and sculptors enrolled in seven art schools.
Now,
The given data is:

So,
From the given data,
We can observe that
The variable that will be on the x-axis (Independent variable) is: Number of painters
The variable that will be on the y-axis (Dependent variable) is: Number of sculptors
But,
From the scatter plot drawn by Jashar,
He interchanged the variables of the axes i.e., he took the independent variable at the y-axis and the dependent variable at the x-axis
Hence, from the above,
We can conclude that the error made by Jashar is the interchange of the variables of the axes

b. Explain the relationship between the number of painters and sculptors enrolled in the art schools.

Answer:
From the given data,
We can observe that for the increase in painters, the number of sculptors also increases
Hence, from the above,
We can conclude that the relationship between the number of painters and sculptors enrolled in the art schools is:
Number of painters ∝ Number of sculptors

c. Reasoning Jashar’s scatter plot shows two possible outliers. Identify them and explain why they are outliers.
Answer:
We know that,
A value that “lies outside” (is much smaller or larger than) most of the other values in a set of data is called an “Outlier”
Now,
The scatter plot for the given data is:

Hence, from the above,
We can conclude that the two possible outliers identified by Jashar are: (11, 6), and (20, 45)

Assessment Practice

Use the scatter plot to answer 10 and 11.

Question 10.
Ten athletes in the Florida Running Club ran two races of the same length. The scatter plot shows their times. Select all statements that are true.

Nine of the times for the first race were at least 16 seconds.
Eight of the times for the second race were less than 17 seconds.
There were seven athletes who were faster in the second race than in the first.
There were three athletes who had the same time in both races.
There were three athletes whose times in the two races differed by exactly 1 second.
Answer:
It is given that
Ten athletes in the Florida Running Club ran two races of the same length. The scatter plot shows their times.
Now,
The given scatter plot is:

Hence,
The correct statements about the given scatter plot is:

Question 11.
What was the greatest difference for a single runner in finishing times in the races?
A. 3 seconds
B. 4 seconds
C. 5 seconds
D. 7 seconds
Answer:
From the given scatter plot,
We can observe that
The lowest time a runner takes for completing a race is approximately 14 seconds
The highest time a runner takes for completing a race is approximately 17 seconds
So,
The greatest difference for a single runner in finishing times in the races is: 3 seconds
Hence, from the above,
We can conclude that option A matches the given situation

Lesson 4.2 Analyze Linear Associations

Solve & Discuss It!

Angus has a big test coming up. Should he stay up and study or go to bed early the night before the test? Defend your recommendation.

Answer:
It is given that
Angus has a big test coming up
Now,
The relationship between the sleeping time and the percentage of marks is also given
Now,
From the given data,
We can observe that
If he went to bed early i.e., at 9:00, then he got 93%
If he studied until 11:00, then he got only 92%
Hence, from the above,
We can conclude that Angus has to go to bed early before the big test

Generalize
Can you make a general statement about which option leads to a better result?
Answer:
Generally, going to bed early will lead to better results before a test

Focus on math practices
Construct Arguments What other factors should Angus also take into consideration to make a decision? Defend your response.
Answer:
The other factors that Angus should also take into consideration when making a decision are:
A) Nature of the exam
B) Coverage of the topics for the particular exam
C) Number of revisions

? Essential Question
How can you describe the association of two data sets?
Answer:
Association (or relationship) between two variables will be described as strong, weak, or none; and the direction of the association may be positive, negative, or none

Try It!
Georgia and her classmates also measured their foot length. Use a pencil to find the trend line. Sketch the trend line for the scatter plot.

Answer:
It is given that
Georgia and her classmates also measured their foot length
Hence,
The representation of a trend line for the given scatter plot is:

Try It!
For each scatter plot, identify the association between the data. If there is no association, state so.
a.

Answer:
The given scatter plot is:

From the above scatter plot,
We can observe that the points are all scattered
Hence, from the above,
We can conclude that the given scatter plot has a weaker association

b.

Answer:
The given scatter plot is:

From the given scatter plot,
The points are all in a non-linear shape
Hence, from the above,
We can conclude that the given scatter plot has a non-linear association
c.

Answer:
The given scatter plot is:

From the scatter plot,
We can observe that the points are all scattered
Hence, from the above,
We can conclude that the given scatter plot has a weaker association

KEY CONCEPT

Scatter plots can show a linear association, a nonlinear association, or no association. For scatter plots that suggest a linear association, you can draw a trend line to show the association. You can assess the strength of the association by looking at the distances of plotted points from the trend line.

Do You Understand?

Question 1.
? Essential Question How can you describe the relationship between the two sets of data?
Answer:
Association (or relationship) between two variables will be described as strong, weak, or none; and the direction of the association may be positive, negative, or none

Question 2.
Look for Relationships How does a trend line describe the strength of the association?
Answer:
The straight line is a trend line, designed to come as close as possible to all the data points. The trend line has a positive slope, which shows a positive relationship between X and Y. The points in the graph are tightly clustered about the trend line due to the strength of the relationship between X and Y.

Question 3.
Construct Arguments How does the scatter plot of a nonlinear association differ from that of a linear association?
Answer:
Scatterplots with a linear pattern have points that seem to generally fall along a line while nonlinear patterns seem to follow along some curve. Whatever the pattern is, we use this to describe the association between the variables.

Do You Know How?

Question 4.
Describe the association between the two sets of data in the scatter plot.

Answer:
The given scatter plot is:

From the given scatter plot,
We can observe that all the points are tightly hugged by a trend line
Hence, from the above,
We can conclude that the given scatter plot has a stronger association

Question 5.
Describe the association between the two sets of data in the scatter plot.

Answer:
The given scatter plot is:

From the given scatter plot,
We can observe that the points are all in a non-linear shape and are closely connected
Hence, from the above,
We can conclude that the given scatter plot has a non-linear association

Practice & Problem Solving

Scan for Multimedia

Question 6.
The scatter plot shows the average heights of children ages 2-12 in a certain country. Which line is the best model of the data?

Answer:
It is given that
The scatter plot shows the average heights of children ages 2-12 in a certain country.
Now,
The best line in the given scatter plot is that line that tightly hugs the maximum points in a scatter plot
Hence, from the above,
We can conclude that line m is the best model of the given data

Question 7.
Does the scatter plot shows a positive, a negative, or no association?

Answer:
The given scatter plot is:

From the given scatter plot,
We can observe that as the value of x increases, the value of y also increases
Hence, from the above,
We can conclude that the given scatter plot has a positive association

Question 8.
Determine whether the scatter plot of the data for the following situation would have a positive or negative linear association.
time working and amount of money earned
Answer:
The given situation is:
Time working and amount of money earned
We know that,
The total amount of work done = Number of days × The amount earned for the work done
Let us suppose the number of days is constant
So,
The total amount of work done ∝ The amount earned for the work done
So,
The more time a person works, the more money that person will earn
Hence, from the above,
We can conclude that the scatter plot of the given data has a positive linear association

Question 9.
Describe the relationship between the data in the scatter plot.

Answer:
The given scatter plot is:

From the given scatter plot,
We can observe that the data in the scatter plot has a decreasing trend with the strong association of data with each other
Hence, from the above
We can conclude that the given scatter plot has a negative linear association

Question 10.
Describe the relationship between the data in the scatter plot.

Answer:
The given scatter plot is:

From the given scatter plot,
We can observe that all the points are in a cyclic fashion
Hence, from the above,
We can conclude that the given scatter plot has a non-linear association

Question 11.
Higher-Order Thinking Describe a real situation that would fit the relationship described.
a. A strong, positive association
Answer:
The real-life examples for a strong, positive association are:
A) The more time you spend running on a treadmill, the more calories you will burn.
B) Taller people have larger shoe sizes and shorter people have smaller shoe sizes.
C) The longer your hair grows, the more shampoo you will need.
D) The less time I spend marketing my business, the fewer new customers I will have.
E) The more hours you spend in direct sunlight, the more severe your sunburn.

b. A strong, negative association
Answer:
The real-life examples for a strong, negative association are:
A) A student who has many absences has a decrease in grades.
B) As the weather gets colder, air conditioning costs decrease.
C) If a train increases speed, the length of time to get to the final point decreases.
D) If a chicken increases in age, the number of eggs it produces decreases.
E) If the sun shines more, a house with solar panels requires less use of other electricity.

Question 12.
A sociologist is studying how sleep affects the amount of money a person spends. The scatter plot shows the results of the study. What type of association does it show between the amount of sleep and money spent?

Answer:
It is given that
A sociologist is studying how sleep affects the amount of money a person spends. The scatter plot shows the results of the study
Now,
From the given scatter plot,
We can observe that the data that is related to the amount of sleep and the amount of money spent is in a cyclic fashion
Hence, from the above,
We can conclude that the association does it show between the amount of sleep and money spent is: Non-linear association

Assessment Practice

Question 13.
Which paired data would likely show a positive association? Select all that apply.
Population and the number of schools
Hair length and shoe size
Number of people who carpool to work and money spent on gas
Hours worked and amount of money earned
Time spent driving and amount of gas in the car
Answer:
We know that,
A positive association is an association that as the value of x increases, the value of y also increases
Hence,
The paired data that would likely show a positive association is

Question 14.
Which paired data would likely show a negative association? Select all that apply.
Population and the number of schools
Hair length and shoe size
Number of people who carpool to work and money spent on gas
Hours worked and amount of money earned
Time spent driving and amount of gas in the car
Answer:
We know that,
A negative association is an association that as the value of x increases, the value of y also decreases
Hence,
The paired data that would likely show a negative association is:

Lesson 4.3 Use Linear Models to Make Predictions

Solve & Discuss It!

Bao has a new tracking device that he wears when he exercises. It sends data to his computer. How can Bao determine how long he should exercise each day if he wants to burn 5,000 Calories per week?

Answer:
It is given that
Bao has a new tracking device that he wears when he exercises. It sends data to his computer
Now,
It is also given that Bao wants to burn 5,000 calories per week
So,
The number of calories Bao wants to burn per day = \(\frac{5,000}{7}\)
= 714.2 calories
= 714 calories
≅ 720 calories
Now,
From the given scatter plot,
We can observe that
For approximately 720 calories to burn, Bao has to exercise 80 – 90 minutes each day
Hence, from the above,
We can conclude that Bao should exercise 80 – 90 minutes each day if he wants to burn 5,000 Calories per week

Focus on math practices

Reasoning Suppose another set of data were plotted with a trend line passing through (25, 100) and (80, 550). Would this indicate that more or fewer calories were burned per minute? Explain.
Answer:

? Essential Question
How do linear models help you to make a prediction?
Answer:
While linear models do not always accurately represent data, and this occurs when actual data does not clearly show a relationship between its two variables, linear models are helpful in determining the future points of data, the expected points of data, and the highest possible accuracy of data.

Try It!

Assuming the trend shown in the graph continues, use the equation of the trend line to predict average fuel consumption in miles per gallon in 2025.
The equation of the trend line is y = x + . In 2025, the average fuel consumption is predicted to be about mpg.

Answer:
The given scatter plot is:

From the scatter plot,
We can observe that
The initial value (y-intercept) is: 15
Now,
We know that,
The equation of the line in the slope-intercept form is:
y = mx + b
Where,
m is the slope
b is the initial value (or) y-intercept
Now,
To find the slope,
The points from the given scatter plot is: (15, 21), (30, 24)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{24 – 21}{30 – 15}\)
= \(\frac{1}{5}\)
So,
The equation of the line is:
y = 0.20x + 15
Now,
The average fuel consumption in 2025 is:
y = 0.20 (2025 – 1980) + 15
y = 0.20 (45) + 15
y = 9 + 15
y = 24 mpg
Hence, from the above,
We can conclude that
The equation of the trend line is:
y = 0.20x + 15
In 2025, the average fuel consumption is predicted to be about 24 mpg

Convince Me!
Why can you use a linear model to predict the y-value for a given x-value?
Answer:
We can use the regression line to predict values of Y was given values of X. For any given value of X, we go straight up to the line and then move horizontally to the left to find the value of Y. The predicted value of Y is called the predicted value of Y, and is denoted Y’.

Try It!

A smoothie café has the ingredients needed to make 50,000 smoothies on a day when the high temperature is expected to reach 90°F. Should the café employees expect to have enough ingredients for the day’s smoothie sales? Explain.
Answer:

KEY CONCEPT
Scatter plots can be used to make predictions about current or future trends.
Look for the corresponding y-value for a given x-value.

Find the equation of the trend line and find the y-value of a given x-value.

Do You Understand?

Question 1.
?Essential Question How do linear models help you to make a prediction?
Answer:
While linear models do not always accurately represent data, and this occurs when actual data does not clearly show a relationship between its two variables, linear models are helpful in determining the future points of data, the expected points of data, and the highest possible accuracy of data.

Question 2.
Model with Math
How do you find the equation of a linear model when you are given the graph but not given the equation?
Answer:
To simplify what has already been said, the easiest way to find the equation of a line is to look for the x and y-intercepts.
One point will be (x, 0) and the other will be (0, y), where x and y are numerical values.
The slope is simply
m = \(\frac{y}{x}\).
When you have the y-intercept, (0, y),
you can use the form
y = mx + b to find the equation for the line.
Consequently, with the notation used, you can represent this as
y=\(\frac{y}{x}\)x + b
where b is the value from (0, y)
x is the value from (x, 0)

Question 3.
Reasoning Can the linear model for a set of data that is presented in a scatter plot always be used to make a prediction about any x-value? Explain.
Answer:
Yes, we can use the linear model to predict values of Y was given values of X. For any given value of X, we go straight up to the line and then move horizontally to the left to find the value of Y. The predicted value of Y is called the predicted value of Y, and is denoted Y’.

Do You Know How?

Question 4.
The graph shows a family’s grocery expenses based on the number of children in the family,

a. Using the slope, predict the difference in the amount spent on groceries between a family with five children and a family with two children.
Answer:
It is given that
The graph shows a family’s grocery expenses based on the number of children in the family,
Now,
The given scatter plot is:

Now,
From the given scatter plot,
The pair that represents the amount spent on groceries in a family with five children is: (5, 175)
The pair that represents the amount spent on groceries in a family with two children is: (2, 140)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{140 – 175}{2 – 5}\)
Slope = \(\frac{-35}{-3}\)
Slope = \(\frac{35}{3}\)
Hence, from the above,
We can conclude that using the slope, the difference in the amount spent on groceries between a family with five children and a family with two children is: \(\frac{35}{3}\)

b. How many children can you predict a family has if the amount spent on groceries per week is $169.47?
Answer:
From the given scatter plot,
The trend line equation is:
y = 21.08x + 85.15
Now,
It is given that the amount spent on groceries per week is $169.47
So,
169.47 = 21.08x + 85.15
21.08x = 169.47 – 85.15
21.08x = 84.32
x = \(\frac{84.32}{21.08}\)
x = 4
Hence, from the above,
We can predict 4 children in a family if he has the amount spent on groceries per week is $169.47

Practice & Problem Solving

Question 5.
Leveled Practice The scatter plot shows the number of people at a fair based on the outside temperature. How many fewer people would be predicted to be at the fair on a 100°F day than on a 75°F day?
The slope is
For each degree that the outside temperature increases, the fair attendance decreases by thousand people.

The difference between 75°F and 100°F is °F.
-0.16 . =
About thousand fewer people are predicted to be at the fair on a 100°F day than on a 75°F day.
Answer:
It is given that
The scatter plot shows the number of people at a fair based on the outside temperature
Now,
The given scatter plot is:

Now,
We know that,
The equation of the trend line that is passing through two points is:
y = mx + b
Where,
m is the slope
b is the initial value (or) y-intercept
Now,
To find the slope,
The given points are: (75, 10K), (100, 6K)
Where,
K is 1000
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{6K – 10K}{100 – 75}\)
= –\(\frac{4,000}{25}\)
= -160
So,
The equation of the trend line is:
y = -160x + b
Substitute (100, 6K) in the above equation
So,
6K = -160 (100) + b
6K + 16K = b
b = 22K
So,
The equation of the trend line is:
y = -160x + 22,000
Now,
At 75° F,
y = -160 (75) + 22,000
y = 10,000
At 100° F,
y = -160 (100) + 22,000
y = 6,000
Hence,
The difference of the people between 75° F and 100° F = 10,000 – 6,000
= 4,000
The difference between 100° F and 75° F = 25° F

Question 6.
Make Sense and Persevere If x represents the number of years since 2000 and y represents the gas price, predict what the difference between the gas prices in 2013 and 2001 is? Round to the nearest hundredth.

Answer:
It is given that
x represents the number of years since 2000 and y represents the gas price,
Now,
We know that,
The equation of the trend line in the slope-intercept form is:
y = mx + b
Where,
m is the slope
b is the y-intercept
Now,
To find the slope of the trend line,
The given points are: (7, 3), (12, 4)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{4 – 3}{12 – 7}\)
Slope = \(\frac{1}{5}\)
So,
y = \(\frac{1}{5}\)x + b
Now,
Substitute (7, 3) or (12, 4) in the above equation
So,
3 = \(\frac{1}{5}\) (7) + b
\(\frac{8}{5}\) = b
So,
The equation of the trend line is:
5y = x + 8
Now,
The gas prices in 2001 is:
5y = 1 + 8
y = \(\frac{9}{5}\)
y = $1.40
The gas prices in 2013 is:
5y = 13 + 8
y = \(\frac{21}{5}\)
y = $4.20
So,
The difference between the gas prices in 2013 and 2001 = $4.20 – $1.40
= $2.80
Hence, from the above,
We can conclude that the difference between the gas prices in 2013 and 2001 is: $2.80

Question 7.
Make Sense and Persevere If x represents the number of months since the beginning of 2016, and y represents the total precipitation to date, predict the amount of precipitation received between the end of March and the end of June.

Answer:
It is given that
x represents the number of months since the beginning of 2016, and y represents the total precipitation to date
Now,
The given scatter plot is:

Now,
From the given scatter plot,
We can observe that the trend line starts from the origin
So,
The equation of the trend line that is passing through the origin is:
y = mx
where,
m is the slope
Now,
To find the slope.
The given points are: (2, 10), (10, 40)
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{40 – 10}{10 – 2}\)
= \(\frac{30}{8}\)
= \(\frac{15}{4}\)
So,
The equation of the trend line is:
y = \(\frac{15}{4}\)x
Now,
At the end of the march,
The amount of precipitation is:
y = \(\frac{15}{4}\) (4)
y = 15 in
At the end of June,
The amount of precipitation is:
y = \(\frac{15}{4}\) (6)
y = \(\frac{45}{2}\)
y = 22.5 inches
So,
The amount of precipitation between the end of March and the end of June = 22.5 – 15
= 7.5 in
Hence, from the above,
We can conclude that the amount of precipitation between the end of March and the end of June is: 7.5 in

Question 8.
The scatter plot shows a hiker’s elevation above sea level over time. The equation of the trend line shown is y = 8.77x + 686. To the nearest whole number, predict what the hiker’s elevation will be after 145 minutes.

Answer:
It is given that
The scatter plot shows a hiker’s elevation above sea level over time.
The equation of the trend line shown is
y = 8.77x + 686.
Where,
8.77 is the slope
686 is the initial value (or) y-intercept
Now,
From the graph,
We can observe that
The x-axis variable – Time
The y-axis variable – Elevation
So,
The hiker’s elevation after 145 minutes is:
y = 8.77 (145) + 686
y = 1,957.65 ft
Hence, from the above,
We can conclude that the hiker’s elevation after 145 minutes will be: 1,957.65 ft

Question 9.
Make Sense and Persevere The graph shows the number of gallons of water in a large tank as it is being filled. Based on the trend line, predict how long it will take to fill the tank with 375 gallons of water.

Answer:
It is given that
The graph shows the number of gallons of water in a large tank as it is being filled
Now,
The given scatter plot is:

Now,
From the given scatter plot,
We can observe that
The initial value (or) y-intercept is: 15
Now,
We know that,
The equation of the trend line that has the initial value is:
y = mx + b
Where,
m is the slope
b is the y-intercept (or) initial value
Now,
To find the slope,
The required points are: (1, 30), (0, 15)
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{15 – 30}{0 – 1}\)
= \(\frac{-15}{-1}\)
= 15
So,
The equation of the trend line is:
y = 15x + 15
Now,
The time taken to fill 375 gallons of water is:
375 = 15x + 15
15x = 375 – 15
15x = 360
x = \(\frac{360}{15}\)
x = 24 minutes
Hence, from the above,
We can conclude that the time taken to fill 375 gallons of water is: 24 minutes

Question 10.
Higher-Order Thinking The graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows. What values could the actual temperature be after the freezer was on for five minutes?

Answer:
It is given that
The graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows.
Now,
From the given scatter plot,
We can observe that,
At 5 minutes of time, the freezer temperature is 15°F
So,
At x = 5, y = 15
But,
According to the given information
At x = 5, y = 15 + 3
So,
y = 18°F
Hence, from the above,
We can conclude that the actual temperature after the freezer was on for five minutes is: 18°F

Assessment Practice

Question 11.
The graph shows the altitude above sea level of a weather balloon over time. The trend line passes through the points (0, 453) and (10, 359). Which statements about the graph are true?
The data show a positive correlation.
The trend line is -9.4x – 453.
In general, the balloon is losing altitude.
The weather balloon started its flight at about 455 feet above sea level.

After 4 minutes, the weather balloon had an altitude of about 415 feet above sea level.
After 395 minutes, the weather balloon had an altitude of about 8 feet above sea level.
Answer:
Let the given options be named as A, B, C, D, E and F respectively
It is given that
The graph shows the altitude above sea level of a weather balloon over time.
The trend line passes through the points (0, 453) and (10, 359)
We know that,
The equation of the trend line that is passing through two points is:
y = mx + b
Where,
m is the slope
b is the initial value (or) y-intercept
We know that
The “y-intercept” is the value of y when x= 0
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{359 – 453}{10 – 0}\)
= \(\frac{-94}{10}\)
= -9.4
So,
The equation of the trend line is:
y = -9.4x + 453
Hence, from the above,
We can conclude that options C, D, and E matches the given situation

Topic 4 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you use a trend line to determine the type of linear association for a scatter plot? Lesson 4.2
Answer:
The straight line is a trend line, designed to come as close as possible to all the data points. The trend line has a positive slope, which shows a positive relationship between X and Y. The points in the graph are tightly clustered about the trend line due to the strength of the relationship between X and Y.

The scatter plot shows the amount of time Adam spent studying and his test scores. Use the scatter plot for Items 2-4.

Question 2.
What relationship do you see between the amount of time spent studying and the test scores? Is the relationship linear? Lesson 4.1
A. In general, Adam scores higher on a test when he spends more time studying. There is not a linear relationship.
B. In general, Adam scores higher on a test when he spends more time studying. There is a positive linear relationship.

C. In general, Adam scores lower on a test when he spends more time studying. There is a negative linear relationship.
D. In general, Adam scores lower on a test when he spends more time studying. There is no relationship.
Answer:
It is given that
The scatter plot shows the amount of time Adam spent studying and his test scores
Now,
The given scatter plot is:

From the given scatter plot,
We can observe that
The association or correlation is positive and there is a linear relationship
Adam is scoring higher on a test when he is studying for more hours
Hence, from the above,
We can conclude that option B matches the given situation

Question 3.
Use the y-intercept and the point (4,90) from the line on the scatter plot. What is the equation of the linear model? Lesson 4.3
Answer:
The given scatter plot is (From Question 2):

Now,
From the given scatter plot,
We can observe that
The initial value (or) y-intercept is: 60
We know that,
The “y-intercept” is the value of y when x = 0
So,
The points required to find the equation of the scatter plot is: (0, 60), (4, 90)
We know that,
The equation of the trend line that has y-intercept is:
y = mx + b
Where,
m is the slope
b is the initial value (or) y-intercept
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{90 – 60}{4 – 0}\)
= \(\frac{30}{4}\)
= \(\frac{15}{2}\)
So,
The equation of the trend line is:
y = \(\frac{15}{2}\)x + 60
Hence, from the above,
We can conclude that the equation of the linear line is:
y = \(\frac{15}{2}\)x + 60

Question 4.
Predict Adam’s test score when he studies for 6 hours. Lesson 4.3
Answer:
We know that,
From the given scatter plot,
The variable on the x-axis is: Time
The variable on the y-axis is: Test scores
Now,
From Problem 3,
The equation of the trend line is:
y = \(\frac{15}{2}\)x + 60
At 6 hours,
y = \(\frac{15}{2}\) (6) + 60
y = 105
Hence, from the above,
We can conclude that Adam’s test score is 105 when he studies for 6 hours

Question 5.
Describe the relationship between the data in the scatter plot. Lesson 4.2

Answer:
The given scatter plot is:

From the above scatter plot,
We can observe that as the value of x increases, the value of y decreases
Hence, from the above,
We can conclude that the given scatter plot has the negative association

Question 6.
The scatter plot shows the mean annual temperature at different elevations. Select all the observations that are true about the scatter plot. Lesson 4.1
The majority of the elevations are in a cluster between 1,250 meters and 2,250 meters.
There is a gap in the data between 500 meters and 1,250 meters.

There is an outlier at about (50, 21).
In general, the mean annual temperature decreases as the elevation increases.
Because there is a gap in the values, there is no association between the temperature and elevation.
Answer:
Let the given options be named as A, B, C, D, and E
It is given that
The scatter plot shows the mean annual temperature at different elevations
Now,
The given scatter plot is:

From the given scatter plot,
We can observe that
There is a cluster between 1,250 m and 2,250m
There is a gap between 500m and 1,250m
In general, the mean annual temperature decreases as the elevation increases.
Because there is a gap in the values, there is no association between the temperature and elevation.
Hence, from the above,
We can conclude that A, B, D, and E matches the given situation
How well did you do on the mid-topic checkpoint? Fill in the stars.

Topic 4 MID-TOPIC PERFORMANCE TASK

A pitcher’s ERA (earned run average) is the average number of earned runs the pitcher allows every 9 innings pitched. The table shows the ERA and the number of wins for starting pitchers in a baseball league.

PART A
Construct a scatter plot of the data in the table.
Answer:
It is given that
A pitcher’s ERA (earned run average) is the average number of earned runs the pitcher allows every 9 innings pitched. The table shows the ERA and the number of wins for starting pitchers in a baseball league.
Now,
The given table is:

Hence,
The representation of the scatter plot for the given data is:

PART B
Identify the association between the data. Explain the relationship between ERA and the number of wins shown in the scatter plot.
Answer:
From the above scatter plot,
We can observe that
As the value of x increases, the value of y decreases
Hence, from the above,
We can conclude that the relationship between ERA and the number of wins as shown in the above scatter plot is a “Negative Correlation”

PART C
Draw a trend line. Write an equation of the linear model. Predict the number of wins of a pitcher with an ERA of 6.
Answer:
We know that,
The equation of the trend line between two points is:
y = mx + b
Where,
m is the slope
b is the initial value (or) y-intercept
Now,
To find the slope,
The points are: (5, 4), (2, 10)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope = y₂ – y₁ / x₂ – x₁
So,
Slope = \(\frac{10 – 4}{2 – 5}\)
= –\(\frac{6}{3}\)
= -2
So,
The equation of the trend line is:
y = -2x + b
Now,
Substitute (5, 4) in the above equation
So,
4 = -2 (5) + b
4 + 10 = b
b = 14
So,
The equation of the trend line is:
y = -2x + 14
Now,
From the given table,
We can observe that
The x-axis variable is: ERA
The y-axis variable is: The number of wins
So,
For x = 6,
y = -2 (6) + 14
y = 14 – 12
y = 2
Hence, from the above,
We can conclude that the number of wins of a pitcher with an ERA of 6 is: 2

Lesson 4.4 Interpret Two-Way Frequency Tables

Explore It!
The owners of a ski resort want to know which is more popular, skiing or snowboarding. The resort conducts a poll, asking visitors their age and which activity they prefer. The results are shown in the table.

A. Use the table to describe the visitors polled.
Answer:
It is given that
The owners of a ski resort want to know which is more popular, skiing or snowboarding. The resort conducts a poll, asking visitors their age and which activity they prefer. The results are shown in the table.
Now,
From the given table,
We can observe that the visiters polled are of the age below 35 and above 35
The activities for which the visitors polled are: Skiing, Snowboarding

B. What information can the owners of the resort determine from the data in the table?
Answer:
From the given table,
We can observe that
The number of visitors that had polled for Skiing and snowboarding
Hence, from the above,
We can conclude that the information the owners of the resort determine from the data in the table is the number of visitors that had polled for Skiing and Snowboarding

C. Make a statement that is supported by the data.
Answer:
The statement that is supported by the given data is:
The number of visitors that are over 35 years is the majority who polled for Skiing

Focus on math practices
Model with Math
How else might you display the data to show the relationship between people’s ages and which activity they prefer?
Answer:
The other way to display the data to show the relationship between people’s ages and the activity they prefer is:

Essential Question
How does a two-way frequency table show the relationships between sets of paired data?
Answer:
Two-way frequency tables are a visual representation of the possible relationships between two sets of categorical data. The categories are labeled at the top and the left side of the table, with the frequency (count) information appearing in the four (or more) interior cells of the table.

Try It!
A weatherman asks 75 people from two different cities if they own rain boots. Complete the two-way frequency table to show the results of the survey.

Answer:
It is given that
A weatherman asks 75 people from two different cities if they own rain boots.
So,
The total number of people who say whether they own rain boots or not are: 75
So,
(The people who say yes + The people who say no)_{City A} + (The people who say yes + the people who say no)_{City B} = 75
Hence,
The complete two-way frequency table that shows the results of the survey is:

Convince Me!
What pattern do you see in the two-way frequency table?
Answer:
Two-way frequency tables are a visual representation of the possible relationships between two sets of categorical data. The categories are labeled at the top and the left side of the table, with the frequency (count) information appearing in the four (or more) interior cells of the table.

Try It!
One hundred students were asked how they traveled to school. Of the girls, 19 rode in a car, 7 rode the bus, and 27 took the train. Of the boys, 12 took the train, 25 rode in a car, and 10 rode the bus. Construct a two-way frequency table. Then tell which mode of transportation is the most popular. Explain.

Answer:
It is given that
One hundred students were asked how they traveled to school. Of the girls, 19 rode in a car, 7 rode the bus, and 27 took the train. Of the boys, 12 took the train, 25 rode in a car, and 10 rode the bus
So,
The complete two-way frequency table for the given situation is:

Now,
From the above, two-way frequency table,
We can observe that more number of people preferred car mode of transportation
Hence, from the above,
We can conclude that the car mode of transportation is the most popular

KEY CONCEPT
A two-way frequency table displays the relationship between paired categorical data. You can interpret the data in the table to draw conclusions.

Do You Understand?

Question 1.
?Essential Question How does a two-way frequency table show the relationship between sets of paired categorical data?
Answer:
Two-way frequency tables are a visual representation of the possible relationships between two sets of categorical data. The categories are labeled at the top and the left side of the table, with the frequency (count) information appearing in the four (or more) interior cells of the table.

Question 2.
Model with Math
How do you decide where to start filling in a two-way frequency table when some of the data are already there?
Answer:
In a two-way frequency table, when there is already data present,
You have to start filling in where there is more data in the table so that all the frequencies can be counted easily and filling of the frequency table will also be fast

Question 3.
Use Structure How can you use the structure of a two-way frequency table to complete it?
Answer:
The steps that are used to complete the two-way frequency table is:
Step 1:
Identify the variables. There are two variables of interest here: the commercial viewed and opinion
Step 2:
Determine the possible values of each variable. For the two variables, we can identify the following possible values
Step 3:
Set up the table
Step 4:
Fill in the frequencies

Do You Know How?

Question 4.
A basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations.

Answer:
It is given that
A basketball coach closely watches the shots of 60 players during basketball tryouts.
So,
Underclassmen + Upperclassmen = 60
Hence,
The complete two-way frequency table that shows the basketball coach’s observations is:

Question 5.
Do the data in the two-way frequency table support the following statement? Explain.
There are more middle school students who wear glasses than high school students who wear contacts.

Answer:
The given statement is:
There are more middle school students who wear glasses than high school students who wear contacts.
Now,
The given two-way frequency table is:

From the given two-way frequency table,
The number of middle school students who wear glasses is: 13
The number of high school students who wear contacts is: 20
So,
The number of middle school students who wear glasses < The number of high school students who wear contacts
Hence, from the above,
We can conclude that the given statement can’t be supported

Practice & Problem Solving

Leveled Practice in 6-8, complete the two-way frequency tables.

Question 6.
You ask 70 of your classmates if they have any siblings. Complete the two-way frequency table to show the results of the survey.

Answer:
It is given that
You ask 70 of your classmates if they have any siblings
So,
(The number of boys and girls who have siblings) + (The number of girls who do not have siblings) = 70
Hence,
The complete two-way frequency table that shows the survey results is:

Question 7.
A company surveyed 200 people and asked which car model they preferred. Complete the two-way frequency table to show the results of the survey.

Answer:
It is given that
A company surveyed 200 people and asked which car model they preferred
So,
(The number of males and females who preferred 2-door car model) + (The number of males and females who preferred 4-door car model) = 200
Hence,
The complete two-way frequency table that shows the results of the survey is:

Question 8.
Make Sense and Persevere
You ask 203 of your classmates how they feel about the school year being made longer. Complete the two-way frequency table to show the results of the survey.

Answer:
It is given that
You ask 203 of your classmates how they feel about the school year being made longer
So,
6th Grade students + 7th Grade students + 8th Grade students = 203
Hence,
The complete two-way frequency table that shows the complete survey results is:

Question 9.
Students at a local school were asked, “About how many hours do you spend on homework each week?” The two-way frequency table shows the results of the survey. Classify the statement below as true or false. Explain.
More students study for 5 to 6 hours than for 1 to 2 hours.

Answer:
It is given that
The two-way frequency table shows the results of the survey.
Now,
The given statement is:
More students study for 5 to 6 hours than for 1 to 2 hours.
Now,
The given two-way frequency table is:

From the given two-way frequency table,
We can observe that
The number of students who study for 5 – 6 hours is: 104
The number of students who study for 1 – 2 hours is: 147
So,
The number of students who study for 1 – 2 hours > The number of students who study for 5 – 6 hours
Hence, from the above,
We can conclude that the given statement is false

Question 10.
Higher-Order Thinking Demi and Margaret record the weather in their respective cities on weekend days over the summer. a. Construct a single, two-way frequency table to show the results.


Answer:
It is given that
Demi and Margaret record the weather in their respective cities on weekend days over the summer
Now,
The given information regarding the given situation is:

Let the struck lines be the number of times that have no rain
Let the non-struck lines be the number of lines that have rain
Hence,
The complete two-way frequency table that shows the results is:

b. Which day saw the least rain? Explain.
Answer:
The condition for the least rain is: The number of times rain occurs + The number of times that no rain occurs
Hence, from the above,
We can conclude that Saturday saw the least rain

Assessment Practice

Question 11.
At one point last year, the local animal shelter had only cats and dogs. There were 74 animals in all. Of the cats, 25 were male and 14 were female. Of the dogs, 23 were male and 12 were female.
PART A
Construct a two-way frequency table of

Answer:
It is given that
At one point last year, the local animal shelter had only cats and dogs. There were 74 animals in all. Of the cats, 25 were male and 14 were female. Of the dogs, 23 were male and 12 were female.
Hence,
The complete two-way table that shows the survey results is:

PART B
For which gender, male or female, is there the data.
a greater need for pet adoption? Explain.
A. There are almost twice as many female pets, so there is a greater need for people to adopt female dogs and cats.
B. There are almost twice as many male pets, so there is a greater need for people to adopt male dogs and cats.
C. There are almost twice as many female pets, so there is a greater need for people to adopt male dogs and cats.
D. There are almost twice as many male pets, so there is a greater need for people to adopt female dogs and cats.
Answer:
From part (A),
the two-way frequency table that matches the given situation is:

So,
From the above two-way frequency table,
We can observe that
There are almost twice as males as females
Hence, from the above,
We can conclude that option D matches the given situation perfectly

Lesson 4.5 Interpret TwoWay Relative Frequency Tables

Solve & Discuss It!

Mr. Day’s math class asked 200 cell phone owners which size phone they prefer. They presented the results in a two-way frequency table. How can you use the data to compare the percent of students who chose the small screen to the percent of adults who chose the small screen?

Answer:
It is given that
Mr. Day’s math class asked 200 cell phone owners which size phone they prefer. They presented the results in a two-way frequency table
Now,
From the given two-way frequency table,
We can observe that
The number of students who chose the small screen is: 48
The number of adults who chose the small screen is: 18
Now,
The percent of students who chose the small screen = \(\frac{The number of small screens chosen by the students}{The total number of screens}\) × 100
= \(\frac{48}{200}\) × 100
= 24%
The percent of adults who chose the small screen = \(\frac{The number of small screens chosen by the adults}{The total number of screens}\) × 100
= \(\frac{18}{200}\) × 100
= 9%
So,
The percent of students who chose the small screen to the percent of adults who chose the small screen
= \(\frac{9}{24}\) × 100
= \(\frac{9 × 100}{24}\)
= 37.5%
Hence, from the above,
We can conclude that the percent of students who chose the small screen to the percent of adults who chose the small screen is: 37.5%

Make Sense and Persevere
How do two-way frequency tables allow you to interpret relationships between categorical data using rows and columns?
Answer:
Two-way frequency tables are a visual representation of the possible relationships between two sets of categorical data. The categories are labeled at the top and the left side of the table, with the frequency (count) information appearing in the four (or more) interior cells of the table.

Focus on math practices
Make Sense and Persevere How does know a percentage change the way you interpret the results?
Answer:
First: work out the difference (increase) between the two numbers you are comparing. Then: divide the increase by the original number and multiply the answer by 100.
So,
% increase = Increase ÷ Original Number × 100.
If your answer is a negative number, then this is a percentage decrease.

? Essential Question
What is the advantage of a two-way relative frequency table for showing relationships between sets of paired data?
Answer:
Two-way relative frequency tables show us percentages rather than counts. They are good for seeing if there is an association between two variables

Try It!
Asha asked 82 classmates whether they play sports on the weekend. The results are shown in the two-way frequency table below.

Convince Me!
How is a two-way relative frequency table different from a two-way frequency table?
Answer:
When a two-way table displays percentages or ratios (called relative frequencies), instead of just frequency counts, the table is referred to as a two-way relative frequency table. These two-way tables can show relative frequencies for the whole table, for rows, or for columns.

Use Asha’s two-way frequency table to complete the two-way relative frequency table.

Answer:
The given two-way table is:

Now,
We know that,
The % of boys or girls who say yes = \(\frac{The number of boys or girls who say yes}{The total number of people}\) × 100
The % of boys or girls who say no = \(\frac{The number of boys or girls who say no}{The total number of people}\) × 100
From the given two-way frequency table,
The total number of people is: 82
Hence,
The complete two-way relative frequency table for the given situation is:

Try It!
Use the data in the table below.

a. How does the percent of students who choose e-books compare to the percent of students who choose audiobooks?
Answer:
From the given two-way relative frequency table,
We can observe that
The % of students who choose e-books is: 52%
The % of students who choose Audiobooks is: 48%
So,
The % of students who choose e-books to the % of students who choose audiobooks
= \(\frac{48}{52}\) × 100
= 92.3%
Hence, from the above,
We can conclude that the % of students who choose e-books to the % of students who choose audiobooks is: 92.3%

b. Is there evidence that 7th graders have a greater tendency to choose audiobooks? Explain.
Answer:
From the given two-way relative frequency table,
The % of 7th-grade students who choose audiobooks is: 58.9%
The % of the 6th-grade students who choose audiobooks is: 36.5%
So,
The % of 7th-grade students who choose audiobooks > The % of 6th-grade students who choose audiobooks
Hence, from the above,
We can conclude that there is a piece of evidence that 7th-graders have a greater tendency to choose audiobooks

KEY CONCEPT
Relative frequency is the ratio of a data value to the total of a row, a column, or the entire data set. It is expressed as a percent. A total two-way relative frequency table gives the percent of the population that is in each group.
In a row two-way relative frequency table, the percents in each row add up to 100%.
In a column two-way relative frequency table, the percents in each column add up to 100%.

Do You Understand?

Question 1.
? Essential Question
What is the advantage of a two-way relative frequency table for showing relationships between sets of paired data?
Answer:
Two-way relative frequency tables show us percentages rather than counts. They are good for seeing if there is an association between two variables

Question 2.
Reasoning when comparing relative frequency by rows or columns only, why do the percentages not total 100%? Explain.
Answer:
When comparing relative frequency by rows or columns only, the individual percentages will not be 100%
So, their total will also not be equal to 100%

Question 3.
Critique Reasoning
Maryann says that if 100 people are surveyed, the frequency table will provide the same information as a total relative frequency table. Do you agree? Explain why or why not.
Answer:
It is given that
Maryann says that if 100 people are surveyed, the frequency table will provide the same information as a total relative frequency table
We know that,
The “Two-way frequency table” gives us information about the categories in the form of counts and frequencies
The “Two-way relative frequency table” gives us information about the categories in terms of percentages of frequencies
Hence, from the above,
We can agree with Maryann

Do You Know How?

In 4-6, use the table. Round to the nearest percent.

Question 4.
What percent of the people surveyed have the artistic ability?
Answer:
From the table,
The number of people who have the artistic ability is: 101
Now,
We know that,
The % of the people surveyed that have the artistic ability = \(\frac{The total number of people who have the artistic ability}{The total number of people}\) × 100
= \(\frac{101}{223}\) × 100
= 45%
Hence, from the above,
We can conclude that 45% of the people surveyed have the artistic ability

Question 5.
What percent of left-handed people surveyed have the artistic ability?
Answer:
From the table,
The number of left-handed people who have artistic ability is: 86
Now,
We know that,
The % of the left-handed people surveyed that have the artistic ability = \(\frac{The total number of left-handed people who have the artistic ability}{The total number of people}\) × 100
= \(\frac{86}{223}\) × 100
= 39%
Hence, from the above,
We can conclude that 39% of the left-handed people surveyed have the artistic ability

Question 6.
What percent of the people who have the artistic ability are left-handed?
Answer:
From the table,
The number of left-handed people who have artistic ability is: 86
Now,
We know that,
The % of the left-handed people surveyed that have the artistic ability = \(\frac{The total number of left-handed people who have the artistic ability}{The total number of people}\) × 100
= \(\frac{86}{223}\) × 100
= 39%
Hence, from the above,
We can conclude that 39% of the left-handed people surveyed have the artistic ability

Practice & Problem Solving

Leveled Practice in 7-8, complete the two-way relative frequency tables.

Question 7.
In a group of 120 people, each person has a dog, a cat, or a bird. The two-way frequency table shows how many people have each kind of pet. Complete the two-way relative frequency table to show the distribution of the data with respect to all 120 people. Round to the nearest tenth of a percent.

Answer:
It is given that
In a group of 120 people, each person has a dog, a cat, or a bird. The two-way frequency table shows how many people have each kind of pet
Hence,
The complete two-way relative frequency table for the given situation is:

Question 8.
There are 55 vehicles in a parking lot. The two-way frequency table shows data about the types and colors of the vehicles. Complete the two-way relative frequency table to show the distribution of the data with respect to color. Round to the nearest tenth of a percent.

Answer:
It is given that
There are 55 vehicles in a parking lot. The two-way frequency table shows data about the types and colors of the vehicles.
Hence,
The completed two-way relative frequency table that shows the distribution of the data with respect to color is:

Question 9.
Men and women are asked what type of car they own. The table shows the relative frequencies with respect to the total population asked. Which type of car is more popular?


Answer:
It is given that
Men and women are asked what type of car they own. The table shows the relative frequencies with respect to the total population asked.
Now,
The given two-way relative frequency table is:

Now,
From the given table,
We can observe that
Most of the people have shown interest in the 4-door type of car
Hence, from the above,
We can conclude that the 4-door type of car is more popular

Question 10.
Make Sense and Persevere Students were asked if they like raspberries. The two-way relative frequency table shows the relative frequencies with respect to the response.

a. What percent of students who do not like raspberries are girls?
Answer:
It is given that
Students were asked if they like raspberries. The two-way relative frequency table shows the relative frequencies with respect to the response.
Now,
From the given two-way frequency table,
We can observe that the % of girls who do not like raspberries are: 48%
Hence, from the above,
We can conclude that the % of students who do not like raspberries are girls is: 48%

b. Is there evidence of an association between the response and the gender? Explain.
Answer:
From the given two-way relative frequency table,
We can observe that
The % of girls who like raspberries is more than the % of boys who like raspberries
The % of girls who do not like raspberries is less than the % of boys who do not like raspberries

Question 11.
Higher-Order Thinking All the workers in a company were asked a survey question. The two-way frequency table shows the responses from the workers in the day shift and night shift.

a. Construct a two-way relative frequency table to show the relative frequencies with respect to the shift.

Answer:
It is given that
All the workers in a company were asked a survey question. The two-way frequency table shows the responses from the workers in the day shift and night shift.
Now,
The given two-way frequency table is:

Hence,
The completed two-way relative frequency table for the survey is:

b. Is there evidence of an association between the response and the shift? Explain.
Answer:
From the two-way frequency table that is mentioned in part (a),
We can observe that
The % of people who opted for the day shift are more than the % of people who opted for the night shift

Assessment Practice

Question 12.
Patients in a blind study were given either Medicine A or Medicine B. The table shows the relative frequencies

Is there evidence that improvement was related to the type of medicine? Explain.
A. The same number of people took each medicine, but the percent of people who reported improvement after taking Medicine B was significantly greater than the percent for Medicine A.
B. The same number of people took each medicine, but the percent of people who reported
improvement after taking Medicine A was significantly greater than the percent for Medicine B.
C. Different numbers of people took each medicine, but the percent of people who reported improvement after taking Medicine B was significantly greater than the percent for Medicine A.
D. Different numbers of people took each medicine, but the percent of people who reported improvement after taking Medicine A was significantly greater than the percent for Medicine B.
Answer:
It is given that
Patients in a blind study were given either Medicine A or Medicine B. The table shows the relative frequencies
We know that,
The number of people will be different
Now,
When we observe the given two-way related frequency table,
The improvement due to Medicine B > The improvement due to Medicine A
Hence, from the above,
We can conclude that option C matches the given situation

3-Act Mathematical Modeling: Reach Out

3-ACT MATH

ACT 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.

Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:

Question 5.
Plot your prediction on the same number line.
Answer:

ACT 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?

Answer:

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math
Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your initial prediction? Explain why.

Answer:

ACT 3

Question 10.
Write the answer you saw in the video.
Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

Reflect

Question 13.
Model with Math
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 14.
Critique Reasoning Choose a classmate’s model. How would you adjust that model?
Answer:

SEQUEL

Question 15.
Model with Math Measure a classmate’s wingspan. Use your model to predict your classmate’s height. How well did your model predicts your classmate’s actual height?

Answer:

Topic 4 REVIEW

? Topic Essential Question

How can you represent the relationship between paired data and use the representation to make predictions?
Answer:
The scatter diagram graphs pairs of numerical data, with one variable on each axis, to look for a relationship between them. If the variables are correlated, the points will fall along a line or curve. The better the correlation, the tighter the points will hug the line

Vocabulary Review

Match each example on the left with the correct word and then provide another example.

Answer:

Use Vocabulary in Writing
Describe the scatter plot at the right. Use vocabulary terms in your description.

Answer:
The given scatter plot is:

Now,
From the given scatter plot,
We can observe that
With the help of measurement data, a trend line is drawn
There is a trend line that is not passing through the origin
There is an outlier present in the given scatter plot

Concepts and Skills Review

LESSON 4.1 Construct and Interpret Scatter Plots

Quick Review
A scatter plot shows the relationship between paired measurement data. Scatter plots can be used to interpret data by looking for clusters, gaps, and outliers.

Practice
The table shows the distance in miles and the price of airfare in dollars.

Question 1.
Construct a scatter plot.

Answer:
It is given that
The table shows the distance in miles and the price of airfare in dollars.
Hence,
The representation of the scatter plot that describes the relationship between the price of airfare and distance is:

Question 2.
Is there a relationship between distance and airfare? Explain.
Answer:
From the above scatter plot,
We can observe that there is no association between distance and airfare
Hence, from the above,
We can conclude that there is no relationship between distance and airfare

LESSON 4.2 Analyze Linear Associations

Quick Review
The association between the data in a scatter plot can be linear or nonlinear. A trend line is a line on a scatter plot, drawn near the points, which approximates the association between paired data. If the data are linear, the association can be positive or negative, and strong or weak.

Practice
Identify the association between the data on each scatter plot.

Question 1.

Answer:
The given scatter plot is:

Now,
From the given scatter plot,
We can observe that as the value of x increases, the value of y decreases
Hence, from the above,
We can conclude that there is a negative association between the data in the scatter plot

Question 2.

Answer:
The given scatter plot is:

From the given scatter plot,
We can observe that the data is in a non-linear trend
Hence, from the above,
We can conclude that there is a non-linear association between the data in the scatter plot

LESSON 4.3 Use Linear Models to Make Predictions

Quick Review
To make predictions, substitute known values into the equation of a linear model to solve for an unknown.

Practice
The scatter plot shows the wages of employees.

Question 1.
If an employee earns $570, what is the expected number of copies sold?
Answer:
It is given that
The scatter plot shows the wages of employees.
Now,
The given scatter plot is:

From the given scatter plot,
We can observe that
The equation of the trend line is:
y = 6x + 120
Where,
y is the wages
x is the number of copies sold
Now,
For y = $570,
570 = 6x + 120
6x = 570 – 120
6x = 450
x = \(\frac{450}{6}\)
x = 75
Hence, from the above,
We can conclude that for an employee wage of $570, the number of copies sold is: 75

Question 2.
If an employee sells 100 copies, what is the expected wage?
Answer:
For x = 100,
y = 6x + 120
y = 6 (100) + 120
y = 600 + 120
y = 720
Hence,f rom the above,
We can conclude that
If an employee sells 100 copies, then the expected wage is: $720

LESSON 4.4 Interpret Two-Way Frequency Tables

Quick Review
A two-way frequency table displays the relationship between paired categorical data.

Practice

Question 1.
The two-way frequency table shows the results of a random survey of movies watched by 100 students. Mrs. Leary said that according to the data, girls are more likely than boys to watch movie A. Is the statement true or false? Explain.

Answer:
It is given that
The two-way frequency table shows the results of a random survey of movies watched by 100 students.
Now,
The given statement is:
Mrs. Leary said that according to the data, girls are more likely than boys to watch movie A.
Now,
The given two-way frequency table is:

Now,
From the given two-way frequency table,
We can observe that
The number of girls who watch movie A > The number of boys who watch movie A
Hence, from the above,
We can conclude that the given statement is true

LESSON 4.5 Interpret Two-Way Relative Frequency Tables

Quick Review
Relative frequency is the ratio of a data value to the total of a row, a column, or the entire data set. It is expressed as a percent.

Practice

The two-way table shows the eye color of 200 cats participating in a cat show.

Question 1.
Make a two-way relative frequency table to show the distribution of the data with respect to gender. Round to the nearest tenth of a percent, as needed.
Answer:
It is given that
The two-way table shows the eye color of 200 cats participating in a cat show.
Now,
The given two-way frequency table is:

Hence,
The representation of the two-way relative frequency table that shows the distribution of the data wrt gender is:

Question 2.
What percent of cats that are female have blue eyes?
Answer:
From the above two-way relative frequency table,
We can observe that there are 30% of cats that are females who have blue eyes
Hence, from the above,
We can conclude that the percent of cats that are females and have blue eyes is: 30%

Topic 4 Fluency Practice

Hidden Clue
For each ordered pair, solve the equation to find the unknown coordinate. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help you answer the riddle below.

A (6, -0.5y + 20 – 0.5y = 13). 6,

B (4 – 3x – 7x = -8, 7) , 7

C (2x + 4 – 6x = 24, 5) , 5

D (5x + 6 – 10x = 31, 1) , 1

E (7x – 3 – 3x = 13, -2) , -2

F (4, -12y + 8y – 21 = -5) 4,

G (44 = 6x – 1 + 9x, –5) 4, , -5

H(-5, 4y + 14 – 2y = 4) -5,

I (-5, 15+ y + 6 + 2y = 0) -5,

J (4, 3y + 32 – y = 18) 4,

K (6, 5y + 20 + 3y = -20) 6,

L (9x – 14 – 8x = -8, -1) , -1

M(-3, -5y + 10 – y = -2) -3,

N(-13 + x – 5 – 4x = -9, 4)
Answer:
The solutions of the above equations are:

Go through the enVision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 3 Use Functions To Model Relationships

Topic 3 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.

Question 1.
The ____ is the ratio of the vertical change to the horizontal change of a line.
Answer:
We know that,
The “Slope” is the ratio of the vertical change to the horizontal change of a line
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Slope”

Question 2.
A relationship that can be modeled by the equation y = mx is a ___
Answer:
We know that,
A relationship that can be modeled by the equation y = mx is a “Proportional relationship”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Proportional relationship”

Question 3.
y-value at which a line of a graph crosses the y-axis is called the ___
Answer:
We know that,
y-value at which a line of a graph crosses the y-axis is called the “y-intercept”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “y-intercept”

Question 4.
An equation written in the form y = mx + b is called the ___
Answer:
We know that,
An equation written in the form y = mx + b is called the “Linear equation” or the “Slope-intercept form”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Linear equation”(or) the “Slope-intercept form”

Slope and y-Intercept

Find the slope and y-intercept of a line that passes through these points.

Question 5.
(2, 2) and (3, 0)
Answer:
The given points are:
(2, 2), and (3, 0)
Compare the given points with (x₁, y₁), (x₂,y₂)
We know that,
Slope(m) = y₂ – y₁ / x₂ – x₁
So,
m = \(\frac{0 – 2}{3 – 2}\)
= \(\frac{-2}{1}\)
= -2
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
We know that,
We can obtain the y-intercept by putting the value of x equal to 0
So,
y = -2x + c
Substitute (3, 0) or (2, 2) in the above equation
So,
0 = -6 + c
So,
c = 6
Hence, from the above,
We can conclude that
The slope of a line that passes through the given points is: -2
The y-intercept of a line is: 6

Question 6.
(1, 5) and (4, 10)
Answer:
The given points are:
(1, 5), and (4, 10)
Compare the given points with (x₁, y₁), (x₂,y₂)
We know that,
Slope(m) = y₂ – y₁ / x₂ – x₁
So,
m = \(\frac{10 – 5}{4 – 1}\)
= \(\frac{5}{3}\)
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
We know that,
We can obtain the y-intercept by putting the value of x equal to 0
So,
y = \(\frac{5}{3}\)x + c
Substitute (4, 10) or (1, 5) in the above equation
So,
5 = \(\frac{5}{3}\) + c
So,
c = \(\frac{10}{3}\)
Hence, from the above,
We can conclude that
The slope of a line that passes through the given points is: \(\frac{5}{3}\)
The y-intercept of a line is: \(\frac{10}{3}\)

Question 7.
(8, 2) and (-8,6)
Answer:
The given points are:
(8, 2), and (-8, 6)
Compare the given points with (x₁, y₁), (x₂,y₂)
We know that,
Slope(m) = y₂ – y₁ / x₂ – x₁
So,
m = \(\frac{6 – 2}{-8 – 8}\)
= \(\frac{4}{-16}\)
= –\(\frac{1}{4}\)
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
We know that,
We can obtain the y-intercept by putting the value of x equal to 0
So,
y = –\(\frac{1}{4}\)x + c
Substitute (-8, 6) or (8, 2) in the above equation
So,
2 = –\(\frac{1}{4}\) (8) + c
So,
c = 4
Hence, from the above,
We can conclude that
The slope of a line that passes through the given points is: –\(\frac{1}{4}\)
The y-intercept of a line is: 4

Compare Proportional Relationships

Jenna’s mother is shopping for energy drinks in 12-ounce bottles for Jenna’s soccer team. Store A sells a case of 18 bottles for $10. Store B sells a case of 12 bottles for $6. Which store sells the drinks for less? Use the graph to compare the unit costs of the drinks.

Question 8.

Answer:
Jenna’s mother is shopping for energy drinks in 12-ounce bottles for Jenna’s soccer team. Store A sells a case of 18 bottles for $10. Store B sells a case of 12 bottles for $6
Now,
The unit cost rate of a bottle in store A = \(\frac{The cost of 18 bottles}{18}\)
= \(\frac{$10}{18}\)
= $0.55
The unit cost rate of a bottle in store B = \(\frac{The cost of 12 bottles}{12}\)
= \(\frac{$6}{12}\)
= $0.5
So,
The representation of the unit cost rate of a bottle in both stores is:

So,
From the above graph,
We can observe that
The unit cost rate of a bottle in store B < The unit cost rate of a bottle in store A
Hence, from the above,
We can conclude that store B sells the drinks for less cost

Linear Equations

Question 9.
Write the equation for the graph of the line shown.

Answer:
The given graph is:

We know that,
The y-intercept is the value of the point that passes through the y-axis
So,
From the graph,
The point that passes through the y-axis is: (0, -6)
So,
The y-intercept is: -6
Now,
To find the slope,
The points are: (2, 2), and (0, -6)
Now,
Compare the given points with (x₁, y₁), (x₂,y₂)
We know that,
Slope(m) = y₂ – y₁ / x₂ – x₁
So,
m = \(\frac{-6 – 2}{0 – 2}\)
= \(\frac{-8}{-2}\)
= 4
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
So,
y = 4x – 6
Hence, from the above,
We can conclude that the equation of the line for the given graph is:
y = 4x – 6

Language Development

Write key words or phrases associated with each representation. Then write function or not a function on the given lines.

Answer:

Topic 3 PICK A PROJECT

PROJECT 3A

What machine could be invented to make your life better?
PROJECT: BUILD A RUBE GOLDBERG MACHINE

PROJECT 3B

What games can you play indoors?
PROJECT: MAKE A MATH
CARD GAME

PROJECT 3C

What are the steps for fixing a leaky pipe?
PROJECT: PLAN A MAINTENANCE ROUTE

PROJECT 3D

If you were to make a video game, what kind of game would it be?
PROJECT: DESIGN A VIDEO GAME ELEMENT

Lesson 3.1 Understand Relations and Functions

Solve & Discuss It!

The 10 members of Photography Club want to raise $500, so they will hold a raffle with donated prizes. Jesse proposes that to reach their goal, each member should sell 50 raffle tickets. Alexis proposes that each member should raise $50.
Whose plan would you recommend? Explain.

RAFFLE TICKETS
$1 1 ticket
$5 6 tickets
$20 25 tickets
Answer:
It is given that
The 10 members of the Photography Club want to raise $500, so they will hold a raffle with donated prizes. Jesse proposes that to reach their goal, each member should sell 50 raffle tickets. Alexis proposes that each member should raise $50.
It is also given that
$1              –       1 ticket
$5              –        6 tickets
$20            –        25 tickets
Now,
According to Jesse’s goal,
Each member should sell 50 raffle tickets to make the total amount of $500
The possible combinations may be:
50 $1 tickets will be sold by each member
Any other combination is not possible
According to Alexis’s plan,
Each member should raise $50 to make the total amount of $500
It is possible and very easy because $50 by each member can be raised in many ways
Hence, from the above,
We can conclude that Alexis’s plan would be recommended

Focus on math practices
Reasoning How are the two plans different? How are they similar?
Answer:
According to Jesse’s goal,
Each member should sell 50 raffle tickets to make the total amount of $500
The possible combinations may be:
50 $1 tickets will be sold by each member
Any other combination is not possible
Now,
According to Alexis’s plan,
Each member should raise $50 to make the total amount of $500
It is possible and very easy because $50 by each member can be raised in many ways

? Essential Question
when is a relation a function?
Answer:
A “Relation” from a set X to a set Y is called a “Function” if and only if each element of X is related to exactly one element in Y.

Try It!
Joe needs to advertise his company. He considers several different brochures of different side lengths and areas. He presents the data as ordered pairs (side length, area).
(4, 24), (5, 35), (8, 24), (2, 20), (9, 27)

Complete the arrow diagram. Is the area of a brochure a function of the side length? Explain.
Answer:
Joe needs to advertise his company. He considers several different brochures of different side lengths and areas. He presents the data as ordered pairs (side length, area).
(4, 24), (5, 35), (8, 24), (2, 20), (9, 27)
We know that,
The ordered pairs can be represented in the form of (x, y)
Where,
x is the input
y is the output
Now,
From the given ordered pairs,
We can observe that for different values of the input, there are different values of output
Note:
If there are the same outputs for the different inputs, then also a relationship is considered a function
So,
The complete arrow diagram for the given ordered pairs are:

Hence, from the above,
We can conclude that the area of the brochure is a function of the side length

Convince Me!
There are two outputs of 24. Does this help you determine whether the relation is a function? Explain.
Answer:
We know that,
If there are the same outputs for the different inputs, then also a relationship is considered a function
Hence, from the above
We can conclude that even for the two outputs of 24, the given relationship is considered a function

Try It!

Frank reverses the ordered pairs to show the heights and ages of the same six students. Is age a function of height? Explain.

Answer:
It is given that
Frank reverses the ordered pairs to show the heights and ages of the same six students.
Now,
The given table is:

Now,
From the given table,
We can observe that
For different values of age, there are different values of height
Where,
Age —-> Input
Height —> Output
Hence, from the above,
We can conclude that age is a function of height

Try It!
Heather claims that she can tell exactly how long a family was at the museum by how much the family pays for parking. Is Heather correct? Explain.
Answer:
It is given that
Heather claims that she can tell exactly how long a family was at the museum by how much the family pays for parking.
Now,
The table for the given situation is: (Example 3)

Now,
From the table,
We can observe that
There are different costs for the different times in hours
So,
We can say that cost is a function of time
Hence, from the above,
We can conclude that Heather’s claim is correct

KEY CONCEPT
A relation is a function if each input corresponds to exactly one output. You can use an arrow diagram or a table to determine whether a relation is a function.
This relation is a function.

This relation is not a function.

Do You Understand?

Question 1.
? Essential Question
when is a relation a function?
Answer:
A relation from a set X to a set Y is called a function if and only if each element of X is related to exactly one element in Y

Question 2.
Model with Math
How can you use different representations of a relation to determine whether the relation is a function?
Answer:
Relations can be displayed as a table, a mapping, or a graph. In a table, the x-values and y-values are listed in separate columns. Each row represents an ordered pair: Displaying a relation as a table

Question 3.
Generalize
Is a relation always a function? Is a function always a relation? Explain.
Answer:
All functions are relations, but not all relations are functions. A function is a relation that for each input, there is only one output. Here are mappings of functions. The domain is the input or the x-value, and the range is the output or the y-value.

Question 4.
Is the relation shown below a function? Explain.

Answer:
The given relation is:

From the given relation,
We can observe that there is the same input for the different outputs,
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is not a function

Question 5.
Is the relation shown below a function? Explain.

Answer:
The given relation is:

From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 6.
Is the relation shown below a function? Explain.
(4,16), (5, 25), (3,9), (6, 36), (2, 4), (1, 1)
Answer:
The given relation is:
(4,16), (5, 25), (3,9), (6, 36), (2, 4), (1, 1)
From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Practice & Problem Solving

Question 7.
The set of ordered pairs (1, 19), (2, 23), (3, 23), (4, 29), (5, 31) represents the number of tickets sold for a fundraiser. The input values represent the day and the output values represent the number of tickets sold on that day.
a. Make an arrow diagram that represents
Answer:
The arrow diagram for the given relation is:

b. is the relation a function? Explain.
Answer:
It is given that
The set of ordered pairs (1, 19), (2, 23), (3, 23), (4, 29), (5, 31) represents the number of tickets sold for a fundraiser. The input values represent the day and the output values represent the number of tickets sold on that day.
Now,
The given relation is:
(1, 19), (2, 23), (3, 23), (4, 29), (5, 31)
From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 8.
Does the relation shown below represent a function? Explain.
(-2, 2), (-7, 1), (-3, 9), (3, 4), (-9,5), (-6, 8)
Answer:
The given relation is:
(-2, 2), (-7, 1), (-3, 9), (3, 4), (-9,5), (-6, 8)
From the given relation,
We can observe that there are different outputs for different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 9.
Is the relation shown in the table a function? Explain.

Answer:
The given relation is:

From the given relation,
We can observe that there are different outputs for the same inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is not a function

Question 10.
Construct Arguments
During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in °C).
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100) Is the relation a function? Explain.

Answer:
It is given that
During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in °C).
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100)
We know that,
An ordered pair can be represented as (x, y)
Where,
x is the time
y is the temperature in °C
Now,
The given relation is:
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100)
From the given relation,
We can observe that there are different outputs for the different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 11.
Reasoning
Taylor has tracked the number of students in his grade since third grade. He records his data in the table below. Is the relation a function? Explain.

Answer:
It is given that
Taylor has tracked the number of students in his grade since third grade. He records his data in the table
Now,
The given table is:

From the given table,
We can observe that there are different outputs (People) for the different inputs (Grade)
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given table is a function

Question 12.
James raises chickens. He tracks the number of eggs his chickens lay at the end of each week. Is this relation a function? Explain.

Answer:
It is given that
James raises chickens. He tracks the number of eggs his chickens lay at the end of each week
Now,
The given relation is:

From the given relation,
We can observe that there are different outputs (Eggs) for the different inputs (Weeks)
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 13.
Relations P and Q are shown below.

a. Make an arrow diagram to represent Relation P.
Answer:
The given relation is:

Hence,
The arrow diagram to represent the relation P is:


b. Make an arrow diagram to represent Relation Q.
Answer:
The given relation is:

Hence,
The arrow diagram to represent the relation Q is:

c. Which relation is a function? Explain.
Answer:
From relation P,
We can observe that there are different outputs for the different inputs
From relation Q,
We can observe that there are different outputs for the same inputs
Hence, from the above,
We can conclude that relation P is a function

Question 14.
Higher-Order Thinking
On a recent test, students had to determine whether the relation represented by the ordered pairs (1, 2), (6, 12), (12, 24), (18, 36) is a function. Bobby drew the arrow diagram on the right and said the relationship was not a function. What error did Bobby most likely make?

Answer:
It is given that
On a recent test, students had to determine whether the relation represented by the ordered pairs (1, 2), (6, 12), (12, 24), (18, 36) is a function. Bobby drew the arrow diagram on the right and said the relationship was not a function.
Now,
From the given arrow diagram and ordered pairs,
We can observe that
In the arrow diagram, inputs and outputs are reversely represented
Hence, from the above,
We can conclude that the error Bobby most likely made is the reversal of inputs and outputs

Assessment Practice

Question 15.
Write the set of ordered pairs that is represented by the arrow diagram at the right. Is the relation a function? Explain.

Answer:
The given arrow diagram is:

In the arrow diagram,
The left side represented the inputs and the right side represented the outputs
So,
The representation of the arrow diagram in the form of the ordered pairs (Input, Output) are:
(49, 13), (61, 36), (10, 27), (76, 52), (23, 52)
From the above relation,
We can observe that there are different outputs for the different inputs
We know that,
A relation can be considered as a function when the different inputs have different outputs
Hence, from the above,
We can conclude that the given relation is a function

Question 16.
Which of these relations are functions? Select all that apply.

Answer:
We know that,
A relation can be considered as a function when the different inputs have different outputs
So,
From the given relations,
Relation 2, Relation 3 are the functions
Hence, from the above,
We can conclude that Relation 2 and Relation 3 are the functions

Lesson 3.2 Connect Representations of Functions

Solve & Discuss It!

Eliza volunteers at a nearby aquarium, where she tracks the migratory patterns of humpback whales from their feeding grounds to their breeding grounds. She recorded the distance, in miles, traveled by the whales each day for the first 7-day period of their migration. Based on Eliza’s data, how long will it take the humpback whales to travel the 3,100 miles to their breeding grounds?

Focus on math practices
Construct Arguments How does finding an average distance the whales travel in miles help with finding a solution to this problem?

? Essential Question
What are different representations of a function?
Answer:
Relationships and functions can be represented as graphs, tables, equations, or verbal descriptions. Each representation gives us certain information. A table of values, mapping diagram, or set of ordered pairs gives us a list of input values and their corresponding output values.

Try It!
As the pump is pumping water, the amount of water in the pool decreases at a constant rate. Complete the statements below. Then graph the function.

The amount of water remaining in the pool is gallons.
The amount of water pumped each hour is gallons.
The equation is
Answer:
It is given that
As the pump is pumping water, the amount of water in the pool decreases at a constant rate
Now,
Let the initial amount of water present in the pool is: 9,000 gallons
So,
The rate of the amount of water that pumped each hour = \(\frac{The initial amount of water present in the pool}{The time that is present where the initial amount of water present}\)
= \(\frac{9,000}{12}\)
= 750 gallons per hour
So,
The amount of water remaining in the pool = The initial amount of water present in the pool – The amount of water that pumped each hour
= 9,000 – 1,500
= 7,500 gallons
Let the number of hours be x
We know that,
The linear equation is in the form of
y = mx + c
So,
The total amount of water present in the pool = The rate at which the water pumps out + The amount of water that pumped each hour
9,000 = 750x + 7,500
Hence, from the above,
We can conclude that
The amount of water remaining in the pool is 7,500 gallons.
The amount of water pumped each hour is 1,500 gallons.
The equation is:
9,000 = 750x + 7,500

Convince Me!
How is the rate of change of this function different from that in Example 1? Explain.
Answer:
The rate of change of the function present in Example 1 is increasing at a constant rate whereas the rate of change of the function in this situation is decreasing at a constant rate

Try It!
Draw a graph that represents a linear function?

Answer:
We know that,
The representation of the linear equation is:
y = mx —–> Slope form
y = mx + c —-> Slope-intercept form
Now,
Let the linear equation in the slope-intercept form be:
y = x + 3
Hence,
The graph of the above linear equation in the coordinate plane is:

KEY CONCEPT

You can represent a function in different ways: in a table, in a graph, or as an equation.
A day at the amusement park costs $10 for an entrance fee and $2.50 for each ride ticket.

Do You Understand?

Question 1.
?Essential Question What are different representations of a function?
Answer:
Relationships and functions can be represented as graphs, tables, equations, or verbal descriptions. Each representation gives us certain information. A table of values, mapping diagram, or set of ordered pairs gives us a list of input values and their corresponding output values.

Question 2.
Use Appropriate Tools How can you use a graph to determine that a relationship is NOT a function?
Answer:
Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is not a function.

Question 3.
Construct Arguments Must the ordered pairs of a function be connected by a straight line or a curve on a graph? Explain.
Answer:
The points can be connected by a straight line. Thus, the ordered pairs represent a linear function.

Do You Know How?

Question 4.
Each week, Darlene tracks the number of party hats her company has in stock. The table shows the weekly stock. Is the relationship a linear function? Use the graph below to support your answer.

Answer:
It is given that
Each week, Darlene tracks the number of party hats her company has in stock. The table shows the weekly stock.
We know that,
A relation is defined as a function only when there are different outputs for different inputs
Now,
From the given table,
We can observe that the outputs (party hats) are different for different inputs (Weeks)
Now,
The given function can be called “Linear function” if the rate of change is constant
The given function can be called a “Non-linear function” if the rate of change is not constant
Now,
The rate of change = Initial value – Next value
Hence, from the above,
We can conclude that the given relationship is a linear function

Question 5.
How can Darlene use the graph above to know when to order more party hats?
Answer:
From the graph,
Darlene know when to order more hats when there is no stack when observing the track sheet

Practice & Problem Solving

Leveled Practice In 6-7, explain whether each graph represents a function.

Question 6.

Answer:
The given graph is:

From the above graph,
We can observe that each input has a different output and the rate of change is constant
Hence, from the above,
We can conclude that the given graph represents a function

Question 7.

Answer:
The given graph is:

From the given graph,
We can observe that each input has a different output but the rate of change is not constant
Hence, from the above,
We can conclude that the given graph represents a function but a non-linear function

Question 8.
Hannah approximates the areas of circles using the equation A = 3r² and records areas of circles with different radius lengths in a table.

a. Graph the ordered pairs from the table.
Answer:
The given table is:

So,
From the table,
The representation of the ordered pairs (in, in²) are:
(1, 3), (2, 12), (3, 27), (4, 48), (5, 75)
Hence,
The representation of the ordered pairs in the coordinate plane is:

b. Is the relation a function? Explain.
Answer:
From part (a),
We can observe that there are different outputs for different inputs
Hence, from the above,
We can conclude that the given relation is a function

Question 9.
Model with Math
The relationship between the number of hexagons, x, and the perimeter of the figure they form, y, is shown in the graph. is the perimeter of the figure a function of the number of hexagons? Explain.

Answer:
It is given that
The relationship between the number of hexagons, x, and the perimeter of the figure they form, y, is shown in the graph
Now,
From the graph,
We can observe that for different values of perimeters, the number of hexagons is also different
Hence, from the above,
We can conclude that the perimeter of a figure is a function of the number of hexagons

Question 10.
Construct Arguments Do the ordered pairs plotted in the graph below represent a function? Explain.

Answer:
The given graph is:

From the given graph,
We can observe that for the different values of x, there are different values of y
Hence, from the above,
We can conclude that the given ordered pairs in the graph represents a function

Question 11.
A train leaves the station at time t = 0. Traveling at a constant speed, the train travels 360 kilometers in 3 hours.
a. Write a function that relates the distance traveled, d, to the time, t.

Answer:
It is given that
A train leaves the station at time t = 0. Traveling at a constant speed, the train travels 360 kilometers in 3 hours.
We know that,
Speed = \(\frac{Distance}{Time}\)
Here,
Speed is constant
So,
Distance = m (Time)
d = mt
Where,
m is the rate of change or proportionality constant
Now,
m = \(\frac{Distance}{Time}\)
= \(\frac{360}{3}\)
= 120 km / hour
Hence, from the above,
We can conclude that the function that relates to the distance d, and time t is:
d = 120t

b. Graph the function and tell whether it is a linear function or a nonlinear function.
The function is a function.
Answer:
From part (a),
The function that relates the distance d, and time t is:
d = 120t
Now,
Compare the above equation with y = mx
Hence,
The graph of the above function is:

Hence, from the above,
We can conclude that the given function is a linear function from the above graph

Question 12.
Higher-Order Thinking Tell whether each graph is a function and justify your answer. Which graph is not a good representation of a real-world situation? Explain.

Answer:
The given graphs are:

From graph A,
We can observe that there are different outputs for the same input
So,
Graph A does not represent the function
From graph B,
We can observe that there are different outputs for the different inputs
So,
Graph B does represent a function
Hence, from the above,
We can say that graph A does not represent the real-world situation

Assessment Practice

Question 13.
You have an ant farm with 22 ants. The population of ants on your farm doubles every 3 months.
PART A
Complete the table.

Answer:
It is given that
You have an ant farm with 22 ants. The population of ants on your farm doubles every 3 months.
Hence,
The completed table for the given situation is:

PART B
Is the relation a function? If so, is it a linear function or a nonlinear function? Explain.

Answer:
From part (a),
We can observe that the ant population is different for the different number of months
Now,
Rate of change = \(\frac{44}{22}\)
= 2
So,
The rate of change is also constant for all the table
Hence, from the above,
We can conclude that the given relation is a linear function

Question 14.
Use the function y = \(\frac{3}{2}\)x + 3 to complete the table of values.

Answer:
The given function is:
y = \(\frac{3}{2}\)x + 3
Hence,
The completed table for the given values of y is:

Lesson 3.3 Compare Linear and Nonlinear Functions

Solve & Discuss It!

Two streaming video subscription services offer family plans with different monthly costs, as shown in the ads below. What do the two plans have in common? How are they different? When is Movies4You a better deal than Family Stream?

Answer:
It is given that
Two streaming video subscription services offer family plans with different monthly costs, as shown in the ads
Now,
Let the number of devices be x
Let the total subscription cost be y
So,
For Movies 4 U,
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
y = $10 + $2x
For Family Stream,
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
y = \(\frac{$12}{4}\) + $1x
y = $3 + $1x
Now,
The above equations are in the form of slope-intercept form
We know that,
The slope-intercept form of the equation is:
y = mx + c
Now,
The common points in the two plans are:
A) The subscription cost of the first device plan
B) The additional fee plan
The different points in the two plans are:
A) The first plan consists of the additional fee of each device
B) The second plan consists of the additional fee for greater than 5 devices
Now,
Movies 4 U is better than Family Stream when the subscription cost of the first device will be less

Model with Math
How can you represent the relationship between cost and number of devices?
Answer:
The representation of the relationship between the cost and the number of devices is:
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices

Focus on math practices
Look for Relationships Describe the relationship between the cost and the number of devices for each service. What do you notice about each relationship?
Answer:
The relationship between the cost and the number of devices for each service is:
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
In the service of Family Stream,
The subscription fee is given for up to 4 devices and the additional subscription fee is for greater than 5 devices

? Essential Question
How can you compare two functions?
Answer:
The two functions can be compared by:
A) Identify the rate of change for the first function
B) Identify the rate of change for the second function
C) Identify the y-intercept of the first function
D) Identify the y-intercept of the second function
E) Compare the properties of each function

Try It!
The welding rate of a third robot is represented by the equation t = 10.8w, where t represents the time in minutes and w represents the number of welding tasks. How does it compare to the other two?
Answer:
It is given that
The welding rate of a third robot is represented by the equation t = 10.8w, where t represents the time in minutes and w represents the number of welding tasks (Refer to Example 1)
Now,
For a third robot,
The wielding rate = \(\frac{The number of wielding tasks (w)}{Time in minutes (t)}\)
= 10.8 (From the equation t = 10.8w)
Now,
When we compare the wielding rates of the three robots,
The wielding rate of the first robot (10.4) < The wielding rate of the third robot (10.8) < The wielding rate of the second robot (11.2)
Hence, from the above,
We can conclude that the comparison of the wielding rates of the three robots is:
The first robot’s wielding rate < The third robot’s wielding rate < The second robot’s wielding rate

Convince Me!
How can linear equations help you compare linear functions?
Answer:
While all linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value).

Try It!
Compare the properties of these two linear functions.

Answer:
The given functions are:

Now,
For function 1,
Find out the rate of change and the y-intercept i.e., the initial value
We know that,
Rate of change = \(\frac{y}{x}\)
Now,
For y = 1 and x = 2.
Rate of change = 0.5
For y = 5.5 and x = 5,
Rate of change = 1.1
SO,
From the above values,
We can say that the rate of change is not constant
So,
The given function is a non-linear function and it does not have any initial value i.e., the y-intercept is 0
Now,
For function 2,
Compare the given equation with
y = mx + c
Where,
m is the slope or the rate of change
c is the initial value or the y-intercept
So,
From the given equation,
Rate of change (m): 2
The y-intercept is: -4
Hence, from the above 2 functions,
We can conclude that
The y-intercept of function1 > The y-intercept of function 2

KEY CONCEPT
You can compare functions in different representations by using the properties of functions.
Compare the constant rate of change and the initial value.

Do You Understand?

Question 1.
? Essential Question
How can you compare two functions?
Answer:
The two functions can be compared by:
A) Identify the rate of change for the first function
B) Identify the rate of change for the second function
C) Identify the y-intercept of the first function
D) Identify the y-intercept of the second function
E) Compare the properties of each function

Question 2.
Reasoning Anne is running on a trail at an average speed of 6 miles per hour beginning at mile marker 4. John is running on the same trail at a constant speed, shown in the table. How can you determine who is running faster?

Answer:
It is given that
Anne is running on a trail at an average speed of 6 miles per hour beginning at mile marker 4. John is running on the same trail at a constant speed, shown in the table.
So,
For Anne,
The rate of change is defined as the average speed
The y-intercept is defined as the beginning point
Hence,
For Anne,
The rate of change is: 6 miles per hour
The y-intercept is: 4
Now,
The given table is:

From the given table,
For John,
Rate of change = \(\frac{y}{x}\) = \(\frac{Mile marker}{Time (hours)}\)
Now,
The total distance traveled by John (y) = Final value – Initial value
= 11.5 – 1
= 10.5 miles
The total time took by John (x) = 1.5 hours
So,
Rate of change = \(\frac{y}{x}\)
= \(\frac{10.5}{1.5}\)
= 7 miles per hour
Now,
We know that,
The y-intercept is the value of y when x = 0
So,
The y-intercept is: 1
Hence, from the above,
By comparing the values of the rate of change,
We can conclude that John is running faster

Question 3.
Reasoning In Item 2, how do Anne and John’s starting positions compare? Explain.
Answer:
We know that,
The starting positions are nothing but the initial positions of both Anne and John i.e., the y-intercepts of both Anne and John
So,
The y-intercept of Anne is: 4
The y-intercept of John is: 1
Hence, from the above,
By comparing the y-intercepts,
We can conclude that
The starting position of Anne > The starting position of John

Do You Know How?

Felipe and Samantha use a payment plan to buy musical instruments. Felipe writes the equation y = -30x + 290 to represent the amount owed, y, after x payments. The graph shows how much Samantha owes after each payment.

Answer:
It is given that
Felipe and Samantha use a payment plan to buy musical instruments. Felipe writes the equation y = -30x + 290 to represent the amount owed, y, after x payments. The graph shows how much Samantha owes after each payment.
Now,
Compare the given equation with
y = mx + c
Where,
m is the rate of change
c is the initial value or the y-intercept
So,
From the given equation
For Felipe,
The rate of change is: -30
The initial value is: 290
Now,
The given graph is:

For Samantha,
From the given graph,
The initial value is: 240
The rate of change = \(\frac{Any value of y}{The value of x that corresponds to the value of y}\)
= \(\frac{120}{6}\)
= 20

Question 4.
Whose musical instrument costs more, Felipe’s or Samantha’s? Explain.
Answer:
We know that,
If the amount owed (y) is less i.e, the rate of change is negative, then the cost of the instrument will also be less
So,
When we compare the rate of change of Felip and Samantha,
The rate of change of Felip < The rate of change of Samantha
Hence, from the above comparison,
We can conclude that the instrument of Samantha costs more

Question 5.
Who will pay more each month? Explain.
Answer:
When we compare the rate of change of Felip and Samantha,
The rate of change of Felip < The rate of change of Samantha
Hence, from the above comparison,
We can conclude that Samantha will pay more each month

Practice & Problem Solving

Question 6.
Two linear functions are shown below. Which function has the greater rate of change?

Answer:
The given linear functions are:

We know that,
The rate of change = \(\frac{y}{x}\)
So,
For Function A,
The rate of change = \(\frac{Any value of y}{The value of x that corresponds to y}\)
= \(\frac{4}{2}\)
= 2
For Function B,
The rate of change = \(\frac{y}{x}\)
=  \(\frac{3}{2}\)
= 1.5
Hence, from the above,
We can conclude that Function A has a greater rate of change

Question 7.
Two linear functions are shown below. Which function has the greater initial value?

Answer:
The given functions are:

We know that,
The initial value is also known as the y-intercept
The y-intercept is the value of y when x = 0
So,
For Function A,
The initial value (y-intercept) is: 4
For function B,
Compare the given equation with
y = mx + c
Where,
m is the rate of change
c is the initial value or the y-intercept
So.
The initial value (y-intercept) is: 3
Hence, from the above,
We can conclude that Function A has the greater initial value

Question 8.
Tell whether each function is linear or nonlinear.

Answer:
The given functions are:

We know that,
To find whether the given function is linear or non-linear, we have to verify whether the rate of change is constant or not
If the rate of change is constant, then the function is linear
If the rate of change is not constant, then the function is non-linear
Now,
For Function A,
The rate of change = \(\frac{y}{x}\)
For x = 1 and y = 2,
The rate of change = 2
For x = 2 and y = 5,
The rate of change = 2.5
Hence,
Function A is a non-linear function
Now,
For Function B,
The rate of change = \(\frac{y}{x}\)
For x = 1 and y = 4,
The rate of change = 4
For x = 1.5 and y = 3,
The rate of change = 2
Hence,
Function B is a non-linear function

Question 9.
Tell whether each function is linear or nonlinear.

Answer:
The given functions are:

We know that,
For a relation to be a graph, each input has a different output but the same input will not have different outputs
Now,
From Function A,
We can observe that there are different inputs for different outputs i.e., the values of x and y are different
So,
The rate of change is not constant since the graph is non-linear
Hence,
Function A is a non-linear function
Now,
From Function B,
Compare the given equation with
y = mx + c
Where,
m = 1
c = 0
So,
The rate of change is constant for Function B
Hence,
Function B is a linear function

Question 10.

Determine whether each function is linear or nonlinear from its graph.

Answer:
The given graphs are:

We know that,
If the graph is a straight line, then the function is a linear function
If the graph is not a straight line, then the function is not a linear function
Hence, from the above,
We can conclude that
The function 1 is a linear function
The function 2 is a non-linear function

Question 11.
Look for Relationships Justin opens a savings account with $4. He saves $2 each week. Does a linear function or a nonlinear function represent this situation? Explain.

Answer:
It is given that
Justin opens a savings account with $4. He saves $2 each week.
Now,
The given table is:

From the given table,
We can observe that there is a constant rate of change
Now,
For weeks,
We can obtain the number of weeks by adding 1 i.e., 0 + 1, 1 + 1, etc
So,
The rate of change is constant i.e., 1
For money in account,
We can obtain the money by adding 2 to the initial amount of money i.e., 4 + 2, 6 + 2, etc
So,
The rate of change is constant i.e., 2
Hence, from the above,
We can conclude that since the rate of change is constant, the given situation represents a linear function

Question 12.
Reasoning The function y = 4x + 3 describes Player A’s scores in a game of trivia, where x is the number of questions answered correctly and y is the score. The function represented in the table shows Player B’s scores. What do the rates of change tell you about how each player earns points?

Answer:
It is given that
The function y = 4x + 3 describes Player A’s scores in a game of trivia, where x is the number of questions answered correctly and y is the score. The function represented in the table shows Player B’s scores.
Now,
For Player A,
The given equation is:
y = 4x + 3
Compare the givene quation with
y = mx + c
Where,
m is the rate of change
c is the y-intercept
So,
The rate of change of Player A is: 4
Now,
The given table is:

For Player B,
From the given table,
We can observe that the rate of change is constant for both the values of x and y
The rate of change for both the values of x and y is: 1
So,
The rate of change of Player B is: 1
So,
The rate of change of Player A > The rate of change of Player B
Hence, from the above,
We can conclude that Player A earns more points

Question 13.
Two athletes are training over a two-week period to increase the number of push-ups each can do consecutively. Athlete A can do 16 push-ups to start, and increases his total by 2 each day. Athlete B’s progress is charted in the table. Compare the initial values for each. What does the initial value mean in this situation?

Answer:
It is given that
Two athletes are training over a two-week period to increase the number of push-ups each can do consecutively. Athlete A can do 16 push-ups to start, and increases his total by 2 each day. Athlete B’s progress is charted in the table
Now,
For Athlete A,
The starting point is: 16
So,
The initial point for Athlete A is: 16
Now,
For Athlete B,
The given table is:

We know that,
The initial point or the y-intercept is the value of y when x = 0
So,
The initial point for Athlete B is: 12
Hence, from the above,
We can conclude that
The initial points in the given situation describe the number of pushups one can do at a time without stop
The initial point for Athlete A > The initial point for Athlete B

Question 14.
Higher-Order Thinking The equation y = 4x – 2 and the table and graph shown at the right describe three different linear functions. Which function has the greatest rate of change? Which has the least? Explain.

Answer:
It is given that
The equation y = 4x – 2 and the table and graph shown at the right describe three different linear functions
Now,
a)
The given equation is:
y = 4x – 2
Compare the given equation with
y = mx + c
Where,
m is the constant rate of change
So,
For the given equation,
The rate of change is: 4
b)
The given table and graph are:

Now,
From the given table,
The rate of change = \(\frac{y}{x}\)
For x = 1 and y = 5,
The rate of change is: 5
For x = 2 and y = 10
The rate of change is: 5
Now,
Since the rate of change is constant for all the cases,
The rate of change for the given table is: 5
Now,
From the given graph,
The given points to find the slope are: (0, 4), and (2, 0)
So,
Slope (or) The rate of change = \(\frac{0 – 4}{2 – 0}\)
= \(\frac{-4}{2}\)
= -2
Now,
When we compare the rate of change for all the three linear functions,
The rate of change of the table > The rate of change of the equation < The rate of change of the graph
Hence, from the above,
We can conclude that
The function that has the greatest rate of change is: Table
The function that has the least rate of change is: Graph

Assessment Practice

Question 15.
The students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time. Which snack did the students eat at a faster rate? Explain.

Answer:
It is given that
The students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time.
Now,
The rate of change of grapes consumption is: 12 grapes per minute
Now,
The given table is:

Now,
The rate of change of carrots consumption = \(\frac{The difference between any 2 values of carrots remaining}{The values of the tie elapsed corresponding to the carrots remaining}\)
= \(\frac{118 – 136}{8 – 6}\)
= –\(\frac{18}{2}\)
= -9 carrots per minute
So,
The consumption rate of grapes > The consumption rate of carrots
Hence, from the baove,
We can conclude that grapes can be eaten at a faster rate

Question 16.
The height of a burning candle can be modeled by a linear function. Candle A has an initial height of 201 millimeters, and its height decreases to 177 millimeters after 4 hours of burning. The height, h, in millimeters, of Candle B, can be modeled by the function h = 290 – 5t, where t is the time in hours. Which of the following statements are true?
The initial height of Candle A is greater than the initial height of Candle B.
The height of Candle A decreases at a faster rate than the height of Candle B.
Candle B will burn out in 58 hours.
After 10 hours, the height of Candle A is 110 millimeters.
Candle A will burn out before Candle B.
Answer:
Let the given options be named as A, B, C, D, and E
It is given that
The height of a burning candle can be modeled by a linear function. Candle A has an initial height of 201 millimeters, and its height decreases to 177 millimeters after 4 hours of burning. The height, h, in millimeters, of Candle B, can be modeled by the function h = 290 – 5t, where t is the time in hours
Now,
The rate of change of Candle A = \(\frac{201 – 177}{4}\)
= \(\frac{24}{4}\)
= 6 millimeters per hour
Now,
For Candle B,
The time to burn the Candle B = \(\frac{290}{5}\)
= 58 hours
Hence, from the above,
We can conclude that B, C, and E match with the given situation

Topic 3 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you determine whether a relation is a function? Lesson 3.1
Answer:
Identify the output values. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

Question 2.
Can an arrow or arrows be drawn from 10.3 so the relation in the diagram is a function? Explain your answer. Lesson 3.1

Answer:
The given diagram is:

For the relation in the diagram to be a function,
The arrows from the right side to the left side can be many but the arrows from the left side to the right side can only be one

Question 3.
Two linear functions are shown below. Which function has the greater rate of change? Justify your response. Lesson 3.3

Answer:
The given linear functions are:

We know that,
For a linear function,
The rate of change is constant
So,
For Function A,
The rate of change = \(\frac{y}{x}\)
= \(\frac{3}{6}\)
= \(\frac{1}{2}\)
= 0.5
For Function B,
The given equation is:
y = \(\frac{1}{2}\)x – 1
y = 0.5x – 1
Compare the above equation with
y = mx + c
Where,
m is the rate of change
So,
The rate of change is: 0.5
Hence, from the above,
We can conclude that the two linear functions have the same rate of change

Question 4.
Neil took 3 math tests this year. The number of hours he spent studying for each test and the corresponding grades he earned is shown in the table. Is the relation of hours of study time to the grade earned on a test a function? Explain why. Use the graph to justify your answer. Lesson 3.2

Answer:
It is given that
Neil took 3 math tests this year. The number of hours he spent studying for each test and the corresponding grades he earned is shown in the table.
Now,
For the number of hours as input and the Grades as the output,
We can observe that
For each value of the hours, there are the same values of the Grade
So,
The relation of hours of study time to the grade earned on a test is not considered a function
Now,
The representation of the relation in the form of the ordered pairs is:
(4, 75), (6, 75), and (6, 82)
Hence,
The representation of the ordered pairs in the coordinate plane is:

Question 5.
Is the function shown linear or nonlinear? Explain your answer. Lesson 3.3

Answer:
The given graph is:

From the given graph,
We can observe that
For each value of x, there is only 1 value of y
So,
The given graph is a function
We know that,
A function is called a linear if the graph is a straight line
A function is called non-linear if the graph is in any shape other than the straight line
Hence, from the above,
We can conclude that the given graph is a non-linear function

How well did you do on the mid-topic checkpoint? Fill in the stars.

Topic 3 MID-TOPIC PERFORMANCE TASK

Sarah, Gene, and Paul are proposing plans for a class fundraiser. Each presents his or her proposal for the amount of money raised, y, for x number of hours worked, in different ways.

Answer:
The given graphs are:

Now,
a)
From the given graph,
We can observe that
For each value of x, there is only 1 value of y
The given graph is a straight line
Now,
The rate of change of the given graph = \(\frac{y}{x}\)
= \(\frac{0 – 10}{1 – 0}\)
= -10
Hence,
The given graph is considered a linear function
b)
From the given table,
To consider a function linear, verify whether the rate of change is constant or not
Now,
For all the values of x,
The rate of change is: 5
For all the values of y,
The rate of change is: 35
So,
The rate of change of the given table = \(\frac{y}{x}\)
= \(\frac{5}{5}\)
= 7
Hence,
The given table is considered a linear function
c)
The given equation is:
y = 10x + 7
Compare the given equation with
y = mx + c
Where,
m is the rate of change
So,
The rate of change for the given equation is: 10
Hence,
The given equation is considered a linear function

PART A
Is each of the proposals represented by linear functions? Explain.
Answer:
Yes, all the proposals are represented by linear functions

PART B
Does the class have any money in the account now? How can you tell?
Answer:
From the given graph,
We can observe that the straight line does not start from 0 but from 10
So,
The initial value of the graph will be: 10
We know that,
The initial value is considered the y-intercept
Hence, from the above,
We can conclude that the class have the money in the account now i.e., $10

PART C
Which fundraising proposal raises money at the fastest rate? Explain.
Answer:
Since the rate of change is the highest for Paul’s proposal,
Paul’s Proposal raises money at the fastest rate

PART D
If Sarah and her classmates are hoping to raise $200, which proposal do you recommend that Sarah and her classmates choose? Explain why you recommend that proposal.
Answer:
It is given that Sarah and her classmates are hoping to raise $200
So,
To raise the money,
We have to choose the plan which has the highest rate of change
Hence, from the above,
We can conclude that Sarah and her classmates choose Paul’s proposal

3-Act Mathematical Modeling: Every Drop Counts

3-ACT MATH

АСТ 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:
After watching the video,
The first question that comes to mind is:
How much amount of water people waste brushing their teeth?

Question 2.
Write the Main Question you will answer.
Answer:
The main question you will answer is:
How much amount of water people waste brushing their teeth?

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.

Answer:
The answer to the main question is: 4 gallons
The prediction of the answer for the main question is according to the surveys done by International Organisations

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:
From the above,
We can observe that
The maximum amount (Too large) of water used to brush teeth is: 4 gallons
The minimum amount (Too small) of water used to brush the teeth is: 2 gallons
Hence,
The representation of the amounts of water used to brush teeth in this situation is:

Question 5.
Plot your prediction on the same number line.
Answer:
From the above,
We can observe that there are minimum and maximum amounts of water used to brush the teeth

Now,
Let x be the amount of water used to brush the teeth
So,
The prediction will be: 2 < x < 4
Hence,
The representation of the prediction on the number line is:

ACT 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?

Answer:
The information in this situation that would be helpful to know is:
How much time did it take to completely brush your teeth?
From the above information,
We can estimate the amount of water used to brush your teeth

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math
Represent the situation using mathematics.
Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.

Answer:

ACT 3

Question 10.
Write the answer you saw in the video.
Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?

Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Answer:

ACT 3

Reflect

Question 13.
Model with Math
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 14.
Be Precise How do the units you chose and the method you used help you communicate your answer?
Answer;

SEQUEL

Question 15.
Use Structure How much water will he save in a year?

Answer:

Lesson 3.4 Construct Functions to Model Linear Relationships

ACTIVITY

Explore It!
Erick wants to buy a new mountain bike that costs $250. He has already saved $120 and plans to save $20 each week from the money he earns for mowing lawns. He thinks he will have saved enough money after seven weeks.

Answer:
It is given that
Erick wants to buy a new mountain bike that costs $250. He has already saved $120 and plans to save $20 each week from the money he earns for mowing lawns. He thinks he will have saved enough money after seven weeks.
So,
The total amount he saved = The amount he saved already + The amount he planned to save each week
Let,
The number of weeks —-> x
The total amount he saved —-> y
So,
y = $20x + $120
Compare the above equation with y = mx + c
Where,
m s the rate of change (or) slope
c is the y-intercept
So,
For the given equation,
The rate of change (m) is: 20
The y-intercept is: 120

A. Complete the table. Then graph the data.

Answer:
The equation is:
y = $20x + $120
So,
The completed table for the above equation is:

So,
The representation of the given equation in the coordinate plane is:

B. How can you tell that the relationship is a linear function from the table? How can you tell from the graph?
Answer:
We know that,
A relation is said to be a function when an input value matches only with an output value
A function is said to be a linear function when the graph of that function is a straight line
So,
From part (a),
From the table,
We can observe that for each value of the week, there is a different amount saved
So,
From the table,
The given relation is said to be a function
From the graph of the equation,
We can observe that the graph is a straight line
So,
We can say that the function is a linear function

Focus on math practices
Generalize How can the different representations help you determine the properties of functions?
Answer:
The different representations of the functions are:
A) Symbolic or Algebraic representation – The properties can be found out by comparing with the standard form
B) Numerical (Tables) representation – The properties can be found out by the rate of change and the initial values
C) Graphical representation – The properties can be found out by the values of x and y
D) Verbal representation – The properties can be found out by the keywords

? Essential Question
How can you use a function to represent a linear relationship?
Answer:
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and c is the initial value of the dependent variable.

Try It!
How will the height of the ramp change if the plan shows that for every 3 inches of height, the triangle should have a base that is 15 inches long?

Graph the function. The slope of the function shown in the graph is . The equation of the function is y =
x. If the base length is 110 inches, then the height of the ramp will be inches.
Answer:
It is given that
The plan shows that for every 3 inches of height, the triangle should have a base that is 15 inches long
So,
The rate of change (m) = \(\frac{Rise}{Run}\)
m = \(\frac{3}{15}\)
m = \(\frac{1}{5}\)
So,
The slope of the function shown in the graph is: \(\frac{1}{5}\)
Now,
We know that,
The representation of the linear equation is:
y = mx
So,
y = \(\frac{1}{5}\)x
Where,
y is the height of the ramp
x is the base length of the triangle
Now,
For the base length of 110 inches,
y = \(\frac{1}{5}\) (110)
y = 22 inches
Hence, from the above,
We can conclude that the height of the ramp will be 22 inches for the base length of 110 inches

Convince Me!
Explain why the initial value and the y-intercept are equivalent.
Answer:
An equation in slope-intercept form of a line includes the slope and the initial value of the function. The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point where the line crosses the y-axis.

Try It!
Jin is tracking how much food he feeds his dogs each week. After 2 weeks, he has used 8\(\frac{1}{2}\) cups of dog food. After 5 weeks, he has used 21\(\frac{1}{4}\) cups. Construct a function in the form y = mx + b to represent the amount of dog food used, y, after x weeks.
Answer:
It is given that
Jin is tracking how much food he feeds his dogs each week. After 2 weeks, he has used 8\(\frac{1}{2}\) cups of dog food. After 5 weeks, he has used 21\(\frac{1}{4}\) cups.
Now,
Let x be the number of weeks
Let y be the number of cups of dog food used
So,
For x = 2, y = 8.5 (The value of 8\(\frac{1}{2}\))
For x = 5, y = 21.25 (The value of 21\(\frac{1}{4}\))
We know that,
The equation in the slope-intercept form is:
y = mx + c
So,
8.5 = 2m + c —–> (1)
21.25 = 5m + c ——> (2)
Solve eq (1) and eq (2)
So,
8.5 = 2m + 21.25 – 5m
8.5 – 21.25 = 2m – 5m
-12.75 = -3m
3m = 12.75
m = \(\frac{12.75}{3}\)
m = 4.25
Now,
For the value of c,
Substitute the value of m either in eq (1) or in eq (2)
So,
8.5 = 2m + c
8.5 = 2 (4.25) + c
8.5 – 8.5 = c
c = 0
Hence, from the above,
We can conclude that the representation of the amount of dog used y, after x weeks in the equation form is:
y = 4.25x

Try It!
The graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing. What function in the form y = mx + b represents this relationship?

Answer:
It is given that
The graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing
Now,
The given graph is:

From the given graph,
We can observe that there is an initial value or y-intercept
So,
The y-intercept from the given graph is: 1
Now,
To find the slope from the given graph,
The given points are: (30, 4), and (10, 2)
So,
Slope (m) = \(\frac{2 – 4}{10 – 30}\)
m = \(\frac{-2}{-20}\)
m = \(\frac{1}{10}\)
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = \(\frac{1}{10}\)x + 1
Hence, from the above,
We can conclude that the equation that represents the given situation is:
y = \(\frac{1}{10}\)x + 1

KEY CONCEPT

A function in the form y= mx + b represents a linear relationship between two quantities, x, and y.

Do You Understand?

Question 1.
?Essential Question How can you use a function to represent a linear relationship?
Answer:
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and c is the initial value of the dependent variable.

Question 2.
Make Sense and Persevere Tonya is looking at a graph that shows a line drawn between two points with a slope of -5. One of the points is smudged and she cannot read it. The points as far as she can tell are (3, 5) and (x, 10). What must the value of x be? Explain.
Answer:
It is given that
Tonya is looking at a graph that shows a line drawn between two points with a slope of -5. One of the points is smudged and she cannot read it. The points as far as she can tell are (3, 5) and (x, 10)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
–5 = \(\frac{10 – 5}{x – 3}\)
-5 = \(\frac{5}{x – 3}\)
-5 (x – 3) = 5
-5 (x) + 5 (3) = 5
-5x + 15 = 5
-5x = 5 – 15
-5x = -10
5x = 10
x = \(\frac{10}{5}\)
x = 2
Hence, from the above,
We can conclude that the value of x is: 2

Question 3.
Reasoning What is the initial value of all linear functions that show a proportional relationship?
Answer:
We know that,
The representation of the proportional relationship is:
y = mx + 0
Where,
m is the slope (or) rate of change
So,
From the above equation,
We can say that the initial value (or) the y-intercept is: 0
Hence, from the above,
We can conclude that the initial value of all linear functions that show a proportional relationship is: 0

Do You Know How?

Question 4.
Write a function in the form y = mx + b for the line that contains the points (-8.3, -5.2) and (6.4, 9.5).
Answer:
The given points are:
(-8.3, -5.2) and (6.4, 9.5)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{9.5 + 5.2}{6.4 + 8.3}\)
m = \(\frac{14.7}{14.7}\)
m = 1
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
-5.2 = -8.3 + c
c = 8.3 – 5.2
c = 3.1
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = x + 3.1

Question 5.
The data in the table below represent a linear relationship. Fill in the missing data.

Answer:
It is given that the data in the table represent a linear relationship
So,
For a linear relationship, for each value of x, there is only 1 value of y
Now,
To find the missing data,
Find the rate of change for x and y
For all the values of x,
The rate of change = 20 – 10
= 10
For all the values of y,
The rate of change = 15 – 10
= 5
Hence,
The completed table with the missing data is:

Question 6.
What is an equation that represents the linear function described by the data in Item 5?
Answer:
From the data in Item 5,
Slope (m) = \(\frac{The rate of change of y}{The rate of change of x}\)
m = \(\frac{5}{10}\)
m = \(\frac{1}{2}\)
We know that,
The equation that represents a linear relationship is:
y = mx
So,
y = \(\frac{1}{2}\)x
Hence, from the above,
The equation that represents the linear function described by the data in Item 5 is:
y = \(\frac{1}{2}\)x

Practice & Problem Solving

Question 7.
A line passes through the points (4, 19) and (9, 24). Write a linear function in the form y = mx + b for this line.
Answer:
The given points are:
(4, 19) and (9, 24)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{24 – 19}{9 – 4}\)
m = \(\frac{5}{5}\)
m = 1
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
19 = 4 + c
c = 19 – 4
c = 15
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = x + 15

Question 8.
What is a linear function in the form y = mx + b for the line passing through (4.5, -4.25) with y-intercept 2.5?
Answer:
It is given that
A line passing through (4.5, -4.25) with y-intercept 2.5
We know that,
The y-intercept is the value of y when x = 0
Now,
The given points are:
(4.5, -4.25) and (0, 2.5)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{2.5 + 4.25}{0 – 4.5}\)
m = \(\frac{6.75}{-4.5}\)
m = -1.5
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = -1.5x + 2.5
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = -1.5x + 2.5

Question 9.
A car moving at a constant speed passes a timing device at t = 0. After 8 seconds, the car has traveled 840 feet. What linear function in the form y = mx + b represents the distance in feet, d, the car has traveled any number of seconds, t, after passing the timing device?
Answer:
It is given that
A car moving at a constant speed passes a timing device at t = 0. After 8 seconds, the car has traveled 840 feet
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
For a constant speed,
Time is considered as input and distance is considered as the output
It is also given that time will start from (0, 0) i..e, at t = 0
So,
The representation of the equation that passes through the origin is:
y = mx
Where,
m is the rate of change (or) slope
In this situation,
The rate of change = \(\frac{Distance}{Time}\)
So,
y = \(\frac{840}{8}\)x
y = 105x
Hence, from the above,
We can conclude that the linear equation that represents the distance traveled by a car after passing the timing device is:
y = 105x

Question 10.
At time t = 0, water begins to drip out of a pipe into an empty bucket. After 56 minutes, 8 inches of water are in the bucket. What linear function in the form y = mx + b represents the amount of water in inches, w, in the bucket after t minutes?
Answer:
It is given that
At time t = 0, water begins to drip out of a pipe into an empty bucket
Now,
In this situation,
Time is considered as output and the amount of water in the bucket is considered as the input
It is also given that time will start from (0, 0) i..e, at t = 0
So,
The representation of the equation that passes through the origin is:
y = mx
Where,
m is the rate of change (or) slope
In this situation,
The rate of change = \(\frac{Time}{The amount of water}\)
So,
y = \(\frac{56}{8}\)x
y = 7x
Hence, from the above,
We can conclude that the linear equation that represents the amount of water in inches, w, in the bucket after t minutes
y = 7x

Question 11.
The graph of the line represents the cost of renting a kayak. Write a linear function in the form y = mx + b to represent the relationship of the total cost, c, of renting a kayak for t hours.

Answer:
It is given that
The graph of the line represents the cost of renting a kayak
Now,
The given graph is:

From the given graph,
We can observe that the line does not pass through the origin and it has the initial value i..e, the y-intercept
So,
From the graph,
The y-intercept is: 8
Now,
To find the slope,
The points from the graph are: (2, 12), and (4, 16)
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{16 – 12}{4 – 2}\)
m = \(\frac{4}{2}\)
m = 2
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = 2x + 8
Hence, from the above,
We can conclude that the relationship to represent the total cost, c, of renting a kayak for t hours is:
y = 2x + 8

Question 12.
An online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.
a. Write a linear function in the form y = mx + b that represents the total cost, y, in dollars, for a single order of x sweatshirts.
Answer:
It is given that
An online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.
So,
The total cost of the sweatshirts = The cost of each sweatshirt + The flat fee of the sweatshirt for shipping
Let the number of sweatshirts be x
let the total cost of the sweatshirts be y
So,
y = $3.99 + $6.50x
Now,
We know that,
The linear equation representation in the slope-intercept form is:
y = mx + c
Hence, from the above,
We can conclude that the equation that represents the total cost, y, in dollars, for a single order of x sweatshirts is:
y = $3.99 + 6.50x

b. Describe how the linear function would change if the shipping charge applied to each sweatshirt.
Answer:
From part (a),
We know that,
The equation that represents the total cost, y, in dollars, for a single order of x sweatshirts, is:
y = $3.99 + $6.50x
Where,
$3.99 is the shipping charge for x sweatshirts,
Now,
If the shipping charge applied to each sweatshirt, then
The representation of the linear equation which we obtained in part (a) is:
y = $6.50x + \(\frac{$3.99}{x}\)
Hence, from the above,
We can conclude that the linear function that represents if the shipping charge applied to each sweatshirt is:
y = $6.50x + \(\frac{$3.99}{x}\)

Question 13.
A store sells packages of comic books with a poster.
a. Model with Math Write a linear function in the form y = mx + b that represents the cost, y, of a package containing any number of comic books, x.
b. Construct Arguments Suppose another store sells a similar package, modeled by a linear function with initial value $7.99. Which store has the better deal? Explain.

Answer:
a)
It is given that a store seller sells packages of comic books with a poster
Now,
Let x be the number of comics
Let y be the amount obtained by selling comics & poster
So,
For x = 6, y = $12.75
For x = 13, y = $19.75
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
To find the slope,
The points are: (6, 12.75), and (13, 19.75)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{19.75 – 12.75}{13 – 6}\)
m = \(\frac{7}{7}\)
m = 1
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
12.75 = 6 + c
c = 12.75 – 6
c = 6.75
Hence, from the above,
We can conclude that the representation of the linear equation that represents the cost, y, of a package containing any number of comic books, x is:
y = x + 6.75
b)
It is given that another store sells a similar package, modeled by a linear function with an initial value of $7.99
Now,
From part (a),
The initial value is: $6.75
From the above initial values,
We can observe that
$6.75 < $7.99
Hence, from the above,
We can conclude that another store has the better deal

Question 14.
Higher-Order Thinking Recommendations for safely thawing a frozen turkey are provided on the packaging.
a. What is the thaw rate of the turkey in hours per pound for refrigerator thawing? For cold water thawing?
b. Write a linear function in the form y = mx + b to represent the time, t, in hours it takes to thaw a turkey in the refrigerator as a function of the weight, w, in pounds of the turkey.

Answer:
a)
We know that,
The thaw rate is nothing but the rate of change
So,
For refrigerator thawing,
Rate of change = \(\frac{The number of pounds}{Time}\)
= 4 pounds per day
We know that
1 day = 24 hours
So,
Rate of change = \(\frac{4}{24}\)
= \(\frac{1}{6}\) pound per hour
For Cold water thawing,
Rate of change = \(\frac{The number of pounds}{Time}\)
= 1 pound per 30 minutes
We know that
1 hour = 60minutes
So,
Rate of change = \(\frac{1 (2)}{1}\)
= 2 pounds per hour
Hence, from the above,
We can conclude that
The rate of change for refrigerator thawing is: \(\frac{1}{6}\) pounds per hour
The rate of change for cold water thawing is: 2 pounds per hour
b)
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
So,
For refrigerator thawing,
The representation of the linear equation as a function of weight w in pounds is:
y = \(\frac{1}{6}\)x + c
For the value of c,
Substitute (24, 4) in the above equation [ The time as x in hours and the weight as y]
So,
4 = \(\frac{24}{6}\) + c
4 = 4 + c
c = 0
Hence, from the above,
We can conclude that the representation of the linear equation as a function of weight w in pounds is:
y = \(\frac{1}{6}\)x

Question 15.
Reasoning The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
a. What is the constant rate of change? What does it represent?

Answer:
It is given that
The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
Now,
The given graph is:

From the given graph,
To find the rate of change,
The points are: (20, 450), and (10, 300)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{300 – 450}{10 – 20}\)
m = \(\frac{150}{10}\)
m = 15
Hence, from the above,
We can conclude that the constant rate of change for the given graph is: 15

b. What is the initial value? What might that represent?
Answer:
We know that,
The initial value is nothing but the y-intercept
Hence, from the above,
We can conclude that the initial value is 50 and this value represents the initial cost of the mulch

Assessment Practice

Question 16.
An international food festival charges for admission and for each sample of food. Admission and 3 samples cost $5.75. Admission and 6 samples cost $8.75. Which linear function represents the cost, y, for any number of samples, x?
A. y = x + 2.75
B. y = 3x + 2.75
C. y = x + 3
D. y = 3x + 3
Answer:
It is given that
An international food festival charges for admission and for each sample of food. Admission and 3 samples cost $5.75. Admission and 6 samples cost $8.75
Now,
Let x be the number of samples
Let y be the cost of samples &Admission
So,
For x = 3, y = $5.75
For x = 6, y = $8.75
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
To find the slope,
The points are: (3, 5.75), and (6, 8.75)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{8.75 – 5.75}{6 – 3}\)
m = \(\frac{3}{3}\)
m = 1
So,
y = x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
5.75 = 3 + c
c = 5.75 – 3
c = 2.75
So,
y = x + 2.75
Hence, from the above,
We can conclude that option A matches the given situation

Question 17.
Some eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.

PART A
Write a linear function in the form y = mx + b that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x.
Answer:
It is given that
Some eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.
Now,
Let x be the number of hours
So,
The total number of muffins students will make = The number of muffins that the students had already made + The number of muffins the students will make in x hours if they made 40 muffins in an hour
y = 40x + 200
We know that,
The representation of the linear equation in the slope-intercept form is:
y = mx + c
Hence, from the above,
We can conclude that
A linear function that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x is:
y = 40x + 200

PART B
How many additional hours would the students spend to make 640 muffins?
Answer:
From part (a),
A linear function that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x is:
y = 40x + 200
It is given that the number of muffins students made is: 640
So,
y = 640
So,
640 = 40x + 200
40x = 640 – 200
40x = 440
4x = 44
x = \(\frac{44}{4}\)
x = 11 hours
Hence, from the above,
We can conclude that the additional hours the students would spend to make 640 muffins is: 11 hours

Lesson 3.5 Intervals of Increase and Decrease

Solve & Discuss It!

Martin will ride his bike from his house to his aunt’s house. He has two different routes he can take. One route goes up and down a hill. The other route avoids the hill by going around the edge of the hill. How do you think the routes will differ? What do you think about the relationship between speed and time?

Answer:
It is given that
Martin will ride his bike from his house to his aunt’s house. He has two different routes he can take. One route goes up and down a hill. The other route avoids the hill by going around the edge of the hill.
We know that,
Speed = \(\frac{Distance}{Time}\)
In this situation,
The distance is constant for the 2 routes
So,
As speed increases, the time decreases
Now,
For route 1:
Route 1 is divided into 2 parts
For the first part (Going up),
As there is friction,
The speed decreases as time increases
For the second part (Going down),
The speed increases as time decreases
For route 2:
Route 2 is divided into 2 parts
For the first part (Going down),
The speed increases as time decreases
For the second part,
The speed is constant as time increases

Focus on math practices
Reasoning How do the characteristics of each route affect Martin’s travel time and speed?
Answer:
We know that,
For the constant distance,
Speed ∝ \(\frac{1}{Time}\)
Now,
Route 1 is in the shape of a parabola
In route 1,
For the first part,
The speed increases as time decreases
For the second part,
The speed decreases as time increases
Now,
Route 2 is increasing for some time and later becomes constant
In route 2,
For the first part,
The speed increases as time decreases
For the second part,
The speed is constant as time increases

? Essential Question
How does a qualitative graph describe the relationship between quantities?
Answer:
The formal term to describe a straight-line graph is linear, whether or not it goes through the origin, and the relationship between the two variables is called a linear relationship. Similarly, the relationship shown by a curved graph is called non-linear.

Try It!
The graph at the right shows another interval in the train’s travel. Which best describes the behavior of the train in the interval shown?
As time , the speed of the train

The function is
Answer:
It is given that the graph shows another interval in the train’s travel
Now,
The given graph is:

From the given graph,
As speed decreases, the time increases
Hence,
The function of the given graph is decreasing in nature

Convince Me!
How would the graph of the function change if the speed of the train was increasing?
Answer:
We know that,
If speed increases, then the time decreases for a constant distance
So,
For a constant distance,
Speed ∝ \(\frac{1}{Time}\)
Hence,
The graph of the function is decreasing in nature as the speed of the train is increased

Try It!
Write a scenario that the graph above could represent. (Example 3)
Answer:
The scenario that the given graph could represent is:
The traveling of a vehicle on a hill

KEY CONCEPT
You can describe the relationship between two quantities by analyzing the behavior of the function relating the quantities in different intervals on a graph.

Do You Understand?

Question 1.
Essential Question How does a qualitative graph describe the relationship between quantities?
Answer:
The formal term to describe a straight-line graph is linear, whether or not it goes through the origin, and the relationship between the two variables is called a linear relationship. Similarly, the relationship shown by a curved graph is called non-linear.

Question 2.
Look for Relationships How would knowing the slope of a linear function help determine whether a function is increasing or decreasing?
Answer:
The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right. For a decreasing function, the slope is negative. The output values decrease as the input values increase.

Question 3.
Use Structure What kind of graph of a function shows the same output values, or y-values, for each input value, or x-value?
Answer:
The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular x value. The y value of a point where a vertical line intersects a graph represents an output for that input x value.

Do You Know How?

Question 4.
What does the graph of the function at each interval represent?

Answer:
The given graph is:

Now,
The given graph is divided into 2 intervals
In the first interval,
The height increases with the increase of time
In the second interval,
The height decreases with the increase of time

Question 5.
In which intervals is the function increasing, decreasing, or constant?

Answer:
The given graph is:

We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that
The intervals that the function is constant – 1, 5
The intervals that the function is increasing – 3, 4
The intervals that the function is decreasing – 2, 6

Practice & Problem Solving

Question 6.
Use the graph to complete the statements.

The function is in intervals 1, 3, and 6.
The function is in intervals 2 and 5.
The function is constant in interval
Answer:
The given graph is:

We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that
The intervals that the function is constant – 4
The intervals that the function is increasing – 1, 3, 6
The intervals that the function is decreasing – 2, 5

Question 7.
The graph below shows the temperature in Paula’s house over time after her mother turned on the air conditioner. Describe the relationship between the two quantities.

Answer:
It is given that
The graph below shows the temperature in Paula’s house over time after her mother turned on the air conditioner
Now,
The given graph is:

From the given graph,
We can observe that
As time increases, the temperature decreases
Hence, from the above,
We can conclude that
Temperature (°F) ∝ \(\frac{1}{Time}\)

Question 8.
You have a device that monitors the voltage across a lamp over time. The results are shown in the graph. Describe the behavior of the function in each interval.
In interval (a), the function is

In the interval (b), the function is
In the interval (c), the function is
In interval (d), the function is
Answer:
It is given that
You have a device that monitors the voltage across a lamp over time. The results are shown in the graph
Now,
The given graph is:

We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that
In interval (a), the function is increasing
In the Interval (b), the function is constant
In the interval (c), the function is decreasing
In the interval (d), the function is constant

Question 9.
The graph below shows the height of a roller coaster over time during a single ride. Circle the intervals in which the function is increasing. In which interval is the increase the greatest?

Answer:
It is given that
The graph below shows the height of a roller coaster over time during a single ride.
Now,
The given graph is:

We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Now,
The given graph with the intervals is:

So,
From the given graph,
The intervals which are increasing in nature are: 1, 3, 5
Hence, from the above,
We can conclude that the increase is the greatest in the 1st interval

Question 10.
Reasoning The graph shows the speed of a car over time. What might the constant intervals in the function represent?

Answer:
It is given that
The graph shows the speed of a car over time
Now,
The given graph is:

We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
We know that,
Speed = \(\frac{Distance}{Time}\)
Now,
From the given graph,
We can observe that
The constant lines represent that the speed becomes constant even though the time is increasing
Hence, from the above,
We can conclude that the constant lines in the given graph represent the constant speed

Question 11.
Higher-Order Thinking A signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
a. In how many intervals is the function decreasing?
b. How are the decreasing intervals alike?

Answer:
It is given that
A signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
Now,
The given graph is:

We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Now,
The given graph with the marked intervals are:

a)
The intervals that the function is decreasing are: 3, 7, 11
b)
The decreasing intervals are all decreasing in nature and have a negative slope

c. How are the decreasing intervals different?
Answer:
The sizes of the decreasing intervals are different
So,
The values of the negative slopes for decreasing intervals will also be different

Question 12.
Critique Reasoning The graph shows the speed of a person riding his stationary exercise bicycle over time. a. A student claims that the function is constant in two intervals. Do you agree? Explain.

Answer:
It is given that
The graph shows the speed of a person riding his stationary exercise bicycle over time. a. A student claims that the function is constant in two intervals.
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
So,
From the given graph,
We can observe that
The number of intervals that are constant in nature is: 4
Hence, from the above,
We can conclude that the claim of the student is not correct

b. What error might the student have made?
Answer:
The student considered only the highest constant intervals but not the lowest intervals
The student also did not consider the last constant interval due to the misconception that it is decreasing in nature but it is constant after that decreasing in nature
So,
The above are the errors the student has made

Question 13.
Look for Relationships The graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.

Answer:
It is given that
The graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.
Now,
The given graph is:

From the given graph,
We can observe that
First, the speed of a roller coaster increases with time, and then it fluctuates between increasing and decreasing as time increases further, and at last, the speed of the roller coaster decreases

Assessment Practice

Question 14.
Which statements about the graph are true? Select all that apply.
The graph is decreasing in intervals (1) and (4).
The graph shows a constant function in interval (2).
The graph is increasing in intervals (2) and (4).
The graph has a constant rate of change.
The graph shows a constant function in interval (3).

Answer:
Let the given options be named as A, B, C, D, and E respectively
Now,
The given graph is:

We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
Hence, from the above,
We can conclude that the options that match with the given situation are: A and E

Lesson 3.6 Sketch Functions from Verbal Descriptions

ACTIVITY

Explain It!

The Environmental Club is learning about oil consumption and energy conservation around the world. Jack says oil consumption in the United States has dropped a lot. Ashley says oil consumption in China is the biggest problem facing the world environment.

A. Do you agree or disagree with Jack’s statement? Construct an argument based on the graph to support your position.
Answer:
The given graph is:

Now,
The statement of Jack is:
Oil consumption in the United States has dropped a lot
Now,
From the given graph,
We can observe that the oil consumption of the United States (The first line in the graph) is constant from 2000 to 2003 and the consumption is constant from 2004 to 2008 and the consumption decreased abruptly from 2008 to 2011
Hence, from the above,
We can conclude that we can agree with Jack’s statement

B. Do you agree or disagree with Ashley’s statement? Construct an argument based on the graph to support your position.
Answer:
The given graph is:

Now,
The statement of Ashley is:
Oil consumption in China is the biggest problem facing the world environment.
Now,
From the given graph,
We can observe that the oil consumption of China (The third line in the graph) is constant from 2000 to 2004 and the consumption is increased abruptly from 2004 to 2008 and the consumption increased minimally from 2008 to 2011
Hence, from the above,
We can conclude that we can agree with Ashley’s statement since China’s consumption of oil will make the other countries suffer

Focus on math practices
Look for Relationships What trend do you see in oil consumption in the United States and in Europe?
Answer:
The given graph is:

From the given graph,
We can observe that
The trend in the oil consumption of the United States and Europe (The first line and the second line in the graph) is:
a) The consumption increases gradually for both the United States and Europe from 2000 to 2003
b) From 2004 to 2008, the oil consumption of the United States becomes constant but the consumption of Europe increases gradually
c) From 2009 to 2011, the oil consumption of the United States and Europe decreases gradually

? Essential Question
How does the sketch of a graph of a function help describe its behavior?

Answer:
From the graph,
We can observe that,
At t = 0,
The level of oxygen is full
At t = 22 minutes,,
The level of oxygen decreases
Hence, from the above,
We can conclude that
The behavior of the time and the oxygen level is:
Time ∝ \(\frac{1}{Oxygen level}\)

Try It!

The weight of the water exerts pressure on a diver. At a depth of 10 feet, the water pressure is 19.1 pounds per square inch (psi) and at a depth of 14 feet, the water pressure is 20.9 psi. Complete the statements, and then sketch the qualitative graph of this function.

The input, or x-variable, is
The output, or y-variable, is
Answer:
It is given that
The weight of the water exerts pressure on a diver. At a depth of 10 feet, the water pressure is 19.1 pounds per square inch (psi) and at a depth of 14 feet, the water pressure is 20.9 psi.
So,
From the above statements,
We can observe that
As the depth of the water increases, the water pressure also increases
So,
The input or x-variable for the given situation is: Depth
The output or y-variable for the given situation is: Water pressure
Now,
To draw the graph,
The required points are: (10, 19.1), and (14, 20.9)
We know that,
the equation of the straight line in the slope-intercept form is:
y = mx + c
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{20.9 – 19.1}{14 – 10}\)
m = \(\frac{1.8}{4}\)
m = \(\frac{9}{20}\)
So,
y = \(\frac{9}{20}\)x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
19.1 = \(\frac{9}{20}\) (10) + c
c = 19.1 – 4.5
c = 14.6
So,
y = \(\frac{9}{20}\)x + 14.6
Hence,
The representation of the above equation in the coordinate plane is:

Convince Me!
Generalize How are the sketches of the two functions similar? How are they different?
Answer:
From Example 1 and the above item’s graph,
We can observe that both graphs have the equation in the slope-intercept form
But,
The graph in Example 1 is decreasing in nature
The graph in the above item is increasing in nature

Try It!
Haru rides his bike from his home for 30 minutes at a fast pace. He stops to rest for 20 minutes and then continues in the same direction at a slower pace for 30 more minutes. Sketch a graph of the relationship of Haru’s distance from home over time.

Answer:
It is given that
Haru rides his bike from his home for 30 minutes at a fast pace. He stops to rest for 20 minutes and then continues in the same direction at a slower pace for 30 more minutes.
Hence,
The graph of the relationship of Haru’s distance from the house over time is:

KEY CONCEPT

You can sketch a graph of a function to describe its behavior. When sketching a function, follow these steps:
1. Identify the two variables (input, output) that have a relationship.
2. Analyze the situation. Look for keywords that indicate that the function is increasing, decreasing, or constant.
3. Sketch the graph.

Do You Understand?

Question 1.
? Essential Question How does the sketch of a graph of a function help describe its behavior?

Answer:
From the graph,
We can observe that,
At t = 0,
The level of oxygen is full
At t = 22 minutes,,
The level of oxygen decreases
Hence, from the above,
We can conclude that
The behavior of the time and the oxygen level is:
Time ∝ \(\frac{1}{Oxygen level}\)

Question 2.
Make Sense and Persevere How do you know which variable goes with which axis when you graph?
Answer:
The independent variable belongs on the x-axis (horizontal line) of the graph and the dependent variable belongs on the y-axis (vertical line).

Question 3.
Reasoning How can you determine the shape of a graph?
Answer:
The four ways to describe shape are whether it is symmetric, how many peaks it has if it is skewed to the left or right, and whether it is uniform. A graph with a single peak is called unimodal. A single peak over the center is called bell-shaped. And, a graph with two peaks is called bimodal.

Do You Know How?

Question 4.
A class plants a tree. Sketch the graph of the height of the tree over time.

a. Identify the two variables.
Answer:
It is given that a class plants a tree
Now,
From the given figure,
We can observe that
As time increases, the height of the tree increases
Hence, from the above,
We can conclude that
Input variable (or) x-coordinate: Time
Output variable (or) y-coordinate: Height of the tree

b. How can you describe the relationship between the two variables?
Answer:
From part (a),
We can observe that
As time increases, the height of the tree increases
Hence, from the above,
We can conclude that the relationship between the variables is:
Time ∝ Height of the tree

c. Sketch the graph.

Answer:
To draw the graph,
The required points are: (0, 3), and (3, 7)
We know that,
the equation of the straight line in the slope-intercept form is:
y = mx + c
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{7 – 3}{3 – 0}\)
m = \(\frac{4}{3}\)
So,
y = \(\frac{4}{3}\)x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
3 = \(\frac{4}{3}\) (0) + c
c = 3 – 0
c = 3
So,
y = \(\frac{4}{3}\)x + 3
Hence,
The representation of the above equation in the coordinate plane is:

Question 5.
An airplane takes 15 minutes to reach its cruising altitude. The plane cruises at that altitude for 90 minutes, and then descend for 20 minutes before it lands. Sketch the graph of the height of the plane over time.

Answer:
It is given that
An airplane takes 15 minutes to reach its cruising altitude. The plane cruises at that altitude for 90 minutes, and then descend for 20 minutes before it lands.
Hence,
The graph of the height of the plane over time is (Height of the plane is any value) is:

Practice & Problem Solving

Question 6.
What relationship between money (in dollars) and time (in months) does this graph show? Write a description of the given graph.

Answer:
The given graph is:

From the given graph,
We can observe that
The amount of money is in an increasing trend for the first half of the month
The amount of money is in a decreasing trend for the second half of the month

Question 7.
When a new laptop became available in a store, the number sold in the first week was high. Sales decreased over the next two weeks and then they remained steady over the next two weeks. The following week, the total number sold by the store increased slightly. Sketch the graph that represents this function over the six weeks.

Answer:
It is given that
When a new laptop became available in a store, the number sold in the first week was high. Sales decreased over the next two weeks and then they remained steady over the next two weeks. The following week, the total number sold by the store increased slightly
Hence,
The graph which represents the given function over the six weeks is:

Question 8.
Aaron’s mother drives to the gas station and fills up her tank. Then she drives to the market. Sketch the graph that shows the relationship between the amount of fuel in the gas tank of her car and time.

Answer:
It is given that
Aaron’s mother drives to the gas station and fills up her tank. Then she drives to the market.
So,
At first, the amount of fuel in the gas tank will be of some initial value and it is in an increasing trend after filling up the tank
After that, when she drives to the market, the amount of fuel will be in a decreasing trend over time
Hence,
The graph that shows the relationship between the amount of fuel in the gas tank of her car and time is:

Question 9.
Melody starts at her house and rides her bike for 10 minutes to a friend’s house. She stays at her friend’s house for 60 minutes. Sketch a graph that represents this description.

Answer:
It is given that
Melody starts at her house and rides her bike for 10 minutes to a friend’s house. She stays at her friend’s house for 60 minutes
Hence,
The graph that represents the above description is:

Question 10.
Which description best represents the graph shown?
A. People are waiting for a train. A train comes and some people get on. The other people wait for the next train. As time goes by, people gradually leave the station.
B. One train arrives and some people get off the train and wait in the station.
C. People are waiting for a train. Everyone gets on the first train that comes.
D. People are waiting for a train. A train comes and some people get on the train. The other people wait for the next train. Another train arrives and all of the remaining people get on.

Answer:
The given graph is:

From the given graph,
We can observe that
At first, there are some people and after some time, the number of people decreased
After some time, there are some people other than the people that decreased and after some further time, the total number of people becomes zero
Hence, from the above,
We can conclude that option D matches the above description

Question 11.
A baker has already made 10 cakes. She can make the same number of cakes each hour, which she does for 5 hours. Sketch the graph of the relationship between the number of cakes made and time.

Answer:
It is given that
A baker has already made 10 cakes. She can make the same number of cakes each hour, which she does for 5 hours.
So,
The initial value for the given graph is: 10
Hence,
The graph of the relationship between the number of cakes made and time is:

Question 12.
Model with Math An air cannon launches a T-shirt upward toward basketball fans. It reaches a maximum height and then descends for a couple seconds until a fan grabs it. Sketch the graph that represents this situation.

Answer:
It is given that
An air cannon launches a T-shirt upward toward basketball fans. It reaches a maximum height and then descends for a couple of seconds until a fan grabs it
We know that,
The projectile (Graph) of an air cannon is always like a “Parabola”
Hence,
The graph that represents the given situation is:

Question 13.
Higher-Order Thinking Write a verbal description of how these two variables are related. The description must suggest at least two intervals. Sketch the graph that represents the verbal description.

Answer:
The verbal description of the relationship between the total people and time in hours is:
People are waiting for a train. A train comes and some people get on the train. The other people wait for the next train. Another train arrives and all of the remaining people get on.
Hence,
The graph that describes the above verbal description is:

Assessment Practice

Question 14.
A baseball team scores the same number of runs in each of the first 4 innings. After that, the team did not score a run for the rest of the game, which lasts 9 innings. Let x represent the innings of the game, and y represent the total number of runs.
PART A
Sketch the graph of this situation below.

Answer:
It is given that
A baseball team scores the same number of runs in each of the first 4 innings. After that, the team did not score a run for the rest of the game, which lasts 9 innings.
Hence,
The graph for the above situation is:

PART B
How would the graph change if the innings in which the team scores runs changes?
Answer:
If the innings in which the team scores runs change, then
The graph will be either in an increasing trend or in a decreasing trend

Topic 3 REVIEW

? Topic Essential Question

How can you use functions to model linear relationships?
Answer:
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and b is the initial value of the dependent variable.

Vocabulary Review

Match each vocabulary term with its definition.

Answer:
Each vocabulary term with its definition is:

Use Vocabulary in Writing

Explain how to write a linear function in the form y = mx + b by using the two points given below. Use vocabulary words in your explanation. (0, -2), (2, 6)
Answer:
The given points are:
(0, -2) and (2, 6)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{6 + 2}{2 – 0}\)
m = \(\frac{8}{2}\)
m = 4
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = 4x + c
Now,
To find the value of c,
Substitute any one of the points in the above equation
So,
-2 = 0 + c
c = -2 – 0
c = -2
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = 4x – 2

Concepts and Skills Review

LESSON 3.1 Understand Relations and Functions

Quick Review
A relation is a set of ordered pairs. A relation is a function if each input, or x-value, has exactly one unique output, or y-value.

Practice

Question 1.
Is the relation shown in the table a function? Explain.

Answer:
The given relation is:

We know that,
A relation is said to be a function if each input has only 1 output
So,
From the given relation,
We can observe that each input has only 1 output
Hence, from the above,
We can conclude that the given relation is a function

Question 2.
Does the relation {(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)} represent a function? Use the arrow diagram. Then explain your answer.

Answer:
The given ordered pairs are:
(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)
We know that,
The ordered pairs are in the form of (input (x), output (y))
In the arrow diagram,
The left side represents the values of the input
The right side represents the values of the output
So,
The representation of the ordered pairs in the arrow diagram is:

We know that,
A relation is said to be a function only when each input corresponds with only 1 output
Now,
From the arrow diagram,
We can observe that
The same input corresponds with 2 outputs
Hence, from the above,
We can conclude that the given relation is not a function

LESSON 3.2 Connect Representations of Functions

Quick Review
You can represent a function in a table, in a graph, or as an equation. The graph of a linear function is a straight line.

Practice
Mark has a $100 gift card to buy apps for his smartphone. Each week, he buys one new app for $4.99.
1. Write an equation that relates the amount left on the card, y, over time, x.
Answer:
It is given that
Mark has a $100 gift card to buy apps for his smartphone. Each week, he buys one new app for $4.99.
Now,
Let y be the amount left on the card
Let x be the number of weeks
So,
The amount left on the card = The total amount of the card – The amount of money he used to buy a new app for x weeks
y = $100 – $4.99x
Hence, from the above,
We can conclude that
The equation that relates the amount left on the card, y, over time, x is:
y = $100 – $4.99x

2. Make a graph of the function.

Answer:
From part (a),
The equation that relates the amount left on the card, y, over time, x is:
y = $100 – $4.99x
Hence,
The representation of the above equation in the coordinate plane is:

LESSON 3.3 Compare Linear and Nonlinear Functions

Quick Review
You can compare functions in different representations by looking at the properties of functions: the constant rate of change and the initial value.

Practice
Two linear functions are shown.

Answer:
The given linear functions are:

Function A:
Compare the given equation with
y = mx + c
Where,
m is the rate of change (or) slope
c is the initial value (or) y-intercept
So,
From the above equation,
The rate of change is: -3
The initial value is: 2
Function B:
From the given table,
We can observe that
The rate of change for all the values of x is: 1
The rate of change for all the values of y is: 2
So,
The rate of change for the table = \(\frac{y}{x}\)
= \(\frac{2}{1}\)
= 2
We know that,
The initial value is the value of y when x = 0
So,
For the given table,
The initial value is: 0
The rate of change is: 2

Question 1.
Which function has the greater initial value? Explain.
Answer:
Function A has the greater initial value

Question 2.
Which function has the greater rate of change?
Answer:
Function B has the greater rate of change

LESSON 3.4 Construct Functions to Model Linear Relationships

Quick Review
A function in the form y = mx + b represents a linear relationship between two quantities, x and y, where m represents the constant rate of change and b represents the initial value.

Practice

Question 1.
What is the equation of a line that passes through (0.5, 4.25) and (2, 18.5) and has a y-intercept of -0.5?
Answer:
It is given that
A line passing through (0.5, 4.25), and (2, 18.5) and has a y-intercept of  -0.5
We know that,
The y-intercept is the value of y when x = 0
Now,
The given points are:
(0.5, 4.25) and (2, 18.5)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Compare the given points with (x₁, y₁), (x₂, y₂)
So,
m = \(\frac{18.5 – 4.25}{2 – 0.5}\)
m = \(\frac{14.25}{1.5}\)
m = 9.5
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = 9.5x – 0.5
Hence, from the above,
We can conclude that the representation of the linear equation for the given points is:
y = 9.5x – 0.5

Question 2.
The graph shows the relationship of the number of gallons being drained from an aquarium over time. What function models the relationship?

Answer:
It is given that
The graph shows the relationship of the number of gallons being drained from an aquarium over time.
Now,
The given graph is:

From the given graph,
We can observe that there is an initial value or y-intercept
So,
The y-intercept from the given graph is: 90
Now,
To find the slope from the given graph,
The given points are: (0, 90), and (9, 0)
So,
Slope (m) = \(\frac{0 – 90}{9 – 0}\)
m = \(\frac{-90}{9}\)
m = -10
We know that,
The representation of the equation in the slope-intercept form is:
y = mx + c
So,
y = -10x + 90
Hence, from the above,
We can conclude that the equation that represents the given relationship is:
y = -10x + 90

LESSON 3-5 Intervals of Increase and Decrease

Quick Review
You can describe the relationship between two quantities by looking at the behavior of the line at different intervals on a qualitative graph. The function is increasing if both x- and y-values increase. The function is decreasing if the y-values decrease as the x-values increase.

Practice

The graph shows the altitude of an airplane over time.

Answer:
We know that,
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes

Question 1.
In which intervals is the graph of the function constant? Explain.
Answer:
The intervals in which the graph of the function is constant are: 2, 4, 6

Question 2.
In which intervals is the graph of the function decreasing? Explain.
Answer:
The intervals in which the graph of the function is decreasing in nature are: 5, 7

LESSON 3.6 Sketch Functions from Verbal Descriptions

Quick Review
You can sketch a graph of a function to describe its behavior. When sketching a function, identify the variables (input, output) that have a relationship, analyze the situation, and then sketch the graph.

Practice

Question 1.
Jack’s mother brings him a bowl of carrots as a snack. At first he does not eat any; then he eats one at a time until half of the carrots are gone. Then he does not eat any more. Sketch a graph that shows the number of carrots in the bowl over time.

Answer:
It is given that
Jack’s mother brings him a bowl of carrots as a snack. At first, he does not eat any; then he eats one at a time until half of the carrots are gone. Then he does not eat anymore
So,
At first,
The number of carrots present in a bowl is: An arbitrary number
The number of carrots Jack consumed is: 0
Hence,
The graph that shows the number of carrots in the bowl over time is:

Topic 3 Fluency Practice

What’s the Message?

In each row, determine which equation has the greater solution. Circle the letter next to the equation with the greater solution in each row.

Answer:
The row of equations that has great solutions are:

What does the zero say to the eight?

Answer:
The statement said by zero to eight is:
You are greater than me

Go through the enVision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 2 Analyze And Solve Linear Equations

Topic Essential Question
How can we analyze connections between linear equations, and use them to solve problems?
Answer:
One of the more obvious “connections” between linear equations is the presence of the same two variables (Generally, in most cases x and y) in these equations.
Assuming that your two equations are distinct (neither is merely a multiple of the other), we can use the “elimination by addition and subtraction” method or “Substitution method” to eliminate one variable, leaving us with an equation in one variable,
solve this 1-variable (x) equation, and then use the resulting value in the other equation to find the value of the other variable (y).
By doing this we find a unique solution (x, y) that satisfies both original equations.
Not only that but also this solution (x, y) will also satisfy both of the original linear equations.

3-ACT MATH

Powering Down
Do you know that feeling when you realize you left your charger at home? Uh-oh. It’s only a matter of time before your device runs out of power. Your battery percentage is dropping, but you still have so much left to do. Think about this during the 3-Act Mathematical Modeling lesson.

Topic 2 enVision STEM Project

Did You Know?
Demography is the study of changes, such as the number of births, deaths, or net migration, occurring in the human population over time.

Deaths Worldwide in 2015 (estimated)
Emigration is the act of leaving one’s country to settle elsewhere. In 2015, 244 million people, or 3.3% of the world’s population, lived outside their country of origin.

Immigration is the act of entering and settling in a foreign country. The United States has the largest immigrant population in the world.

Your Task: Modeling Population Growth

Human population numbers are in constant flux. Suppose a country has a population of 20 million people at the start of one year and during the year there are 600,000 births, 350,000 deaths, 100,000 immigrants, and 5,000 emigrants. You and your classmates will determine the total population at the end of the year and then model the expected change over a longer period.
Answer:
It is given that the population at the start of the year is 20 million people and during that year, there are 600,000 births, 350,000 deaths, 100,000 immigrants, and 5,000 emigrants
So,
The total population at the end of the year = (Total population at the start of the year) – ( Births + Deaths + Immigrants + Emigrants at that year)
= 20 million – (600,000 + 350,000 + 100,000 + 5,000)
= 20 million – 10.5 million
= 9.5 million
Change in Population = (Births + Immigration) – (Deaths + Emigration)
= (600,000 + 100,000) – (350,000 + 5,000)
= 700,000 – 355,000
= 345,000
Hence, from the above,
We can conclude that
The total population at the end of the year is: 9.5 million
The change in population at that year is: 345,000

Topic 2 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.
inverse operations
like terms
proportion
variables

Question 1.
In an algebraic expression, __________ are terms that have the same variables raised to the same exponents.
Answer:
We know that,
In an algebraic expression, “Like terms” are terms that have the same variables raised to the same exponents.
Hence, from the above,
We can conclude that the best term that fits the given definition is: Like terms

Question 2.
Quantities that represent an unknown value are _________.
Answer:
We know that,
Quantities that represent an unknown value are “Variables”
Hence, from the above,
We can conclude that the best term that fits the given definition is: Variables

Question 3.
A _________ is a statement that two ratios are equal.
Answer:
We know that,
A “Proportion” is a statement that two ratios are equal.
Hence, from the above,
We can conclude that the best term that fits the given definition is: Proportion

Question 4.
Operations that “undo” each other are __________.
Answer:
We know that,
Operations that “undo” each other are ” Inverse Operations”
Hence, from the above,
We can conclude that the best term that fits the given definition is: Inverse Operations

Identify Like Terms

Complete the statements to identify the like terms in each expression.
Question 5.
4x + 7y – 62 + 6y – 9x
4x and ______ are like terms.
7y and _______ are like terms.
Answer:
The given expression is:
4x + 7y – 62 + 6y – 9x
We know that,
The “Like terms” are terms that have the same variables raised to the same exponents.
Hence, from the above,
We can conclude that
4x and 9x are like terms
7y and 6y are like terms

Question 6.
\(\frac{1}{2}\)s – (6u – 9u) + \(\frac{1}{10}\)s + 25
\(\frac{1}{2}\)s and _______ are like terms.
6u and _______ are like terms.
Answer:
The given expression is:
\(\frac{1}{2}\)s – (6u – 9u) + \(\frac{1}{10}\)s + 25
= \(\frac{1}{2}\)s + 9u – 6u + \(\frac{1}{10}\)s + 25
We know that,
The “Like terms” are terms that have the same variables raised to the same exponents.
Hence, from the above,
We can conclude that
\(\frac{1}{2}\)s and \(\frac{1}{10}\)s are like terms
6u and 9u are like terms

Solve One-Step Equations

Simplify each equation.
Question 7.
2x = 10
Answer:
The given expression is:
2x = 10
Divide by 2 into both sides
\(\frac{2}{2}\)x = \(\frac{10}{2}\)
x = 5
Hence, from the above,
We can conclude that the value of x is: 5

Question 8.
x + 3 = 12
Answer:
The given expression is:
x + 3 = 12
Subtract with 3 on both sides
x + 3 – 3 = 12 – 3
x = 9
Hence, from the above,
We can conclude that the vaue of x is: 9

Question 9.
x – 7 = 1
Answer:
The given expression is:
x – 7 = 1
Add with 7 on both sides
x – 7 + 7 = 1 + 7
x = 8
Hence, from the above,
We can conclude that the value of x is: 8

Simplify Fractions

Question 10.
Explain how to simplify the fraction \(\frac{12}{36}\).
Answer:
The given fraction is:
\(\frac{12}{36}\)
From the given fraction,
We can observe that the numerator and the denominator are the multiples of 12
So,
Divide the numerator by 12 and the denominator by 12
So,
\(\frac{12}{36}\) = \(\frac{1}{3}\)
Hence,
The simplified form of the given fraction is: \(\frac{1}{3}\)

Language Development
Fill in the Venn diagram to compare and contrast linear equations of the form y = mx and y = x + b.

In the box below, draw graphs to represent each form of the linear equations.

Topic 2 PICK A PROJECT

PROJECT 2A
If you had to escape from a locked room, how would you start?
PROJECT: DESIGN AN ESCAPE-ROOM ADVENTURE

PROJECT 2B
What animal would you most like to play with for an hour? Why?
PROJECT: PLAN A PET CAFÉ

PROJECT 2C
If you wrote a play, what would it be about?
PROJECT: WRITE A PLAY

PROJECT 2D
How many tiny steps does it take to cross a slackline?
PROJECT: GRAPH A WALKING PATTERN

Lesson 2.1 Combine Like Terms to solve Equations

Explore It!
A superintendent orders the new laptops shown below for two schools in her district. She receives a bill for $7,500.

I can… solve equations that have like terms on one side.

A. Draw a representation to show the relationship between the number of laptops and the total cost.
Answer:
It is given that she receives a bill for $7,500
So,
The total cost of the laptops that are given in the above figure = $7,500
Now,
Let the cost of a laptop be $x
So,
$3x + $4x + $3x = $7,500
$10x = $7,500
Hence, from the above,
We can conclude that
The representation to show the relationship between the number of laptops and the total cost is:
$10x = $7,500

B. Use the representation to write an equation that can be used to determine the cost of one laptop.
Answer:
From part (a),
The representation to show the relationship between the number of laptops and the total cost is:
$10x = $7,500
Divide with 10 into both sides
So,
\(\frac{$10x}{10}\) = \(\frac{$7,500}{10}\)
$x = $750
Hence, from the above,
We can conclude that
The representation to write an equation that can be used to determine the cost of one laptop is:
$x = $750

Focus on math practices
Reasoning Why is it important to know that each laptop costs the same amount?
Answer:
From the given figure,
We can observe that all the laptops are of the same type
So,
Each laptop will cost the same amount since all the laptops are the same

Essential Question
How do you solve equations that contain like terms?
Answer:
We will solve the equations that contain like terms by rearranging the like terms on either the left side or the right side

Try It!

Selena spends $53.94 to buy a necklace and bracelet set for each of her friends. Each necklace costs $9.99, and each bracelet costs $7.99. How many necklace and bracelet sets, s, did Selena buy?
Selena buys necklace and bracelet sets for _________ friends.
_____ s + ______ s = 53.94
______ s = 53.94
s = ______
Answer:
Let each necklace and each bracelet be s
It is given that
The cost of each necklace is: $9.99
The cost of each bracelet is: $7.99
The total cost of a necklace and a bracelet is: $53.94
So,
$9.99s + $7.99s = $53.94
$17.98s = $53.94
$1798s = $5394
Divide by 1798 on both sides
\(\frac{$1798}{1798}\)s = \(\frac{$5394}{1798}\)
s = 3
Hence, from the above,
We can conclude that the number of necklace and bracelet sets that Selena buy is: 3

Convince Me!
Suppose the equation is 9.99s + 7.99s + 4.6 = 53.94. Can you combine the s terms and 4.6? Explain.
Answer:
The given equation is:
9.99s + 7.99s + 4.6 = 53.94
We know that,
We can only combine the terms only when they are the “Like terms”
So,
In the given equation,
9.99s and 7.99s are the like terms
53.94 and 4.6 are the like terms
Hence, from the above,
We can conclude that we can not combine the s terms and 4.6

Try It!

Nat’s grocery bill was $150, which included a 5% club discount. What was Nat’s bill before the discount? Write and solve an equation.
Answer:
It is given that Nat’s grocery bill was $150 which included a 5% club discount
Now,
Let x be Nat’s bill before the discount
So,
To find Nat’s bill before discount, we have to find the value of 5% of 150 and add its value from 150
We know that,
The value of the bill will always be less after discount when compared to the value of the bill before discount
Now,
Nat’s bill before the discount = (Nat’s bill which included a 5% club discount) + (Value of 5% of 150)
x = $150 + (\(\frac{5}{100}\) × 150)
x = $150 + \(\frac{5 × 150}{100}\)
x = $150 + \(\frac{750}{100}\)
x = $150 + $7.5
x = $157.5
Hence, from the above,
We can conclude that Nat’s bill before the discount is: $157.5

Try It!

Solve for d.
a. –\(\frac{1}{4}\)d – \(\frac{2}{5}\)d = 39
Answer:
The given expression is:
–\(\frac{1}{4}\)d – \(\frac{2}{5}\)d = 39
-d (\(\frac{1}{4}\) + \(\frac{2}{5}\)) = 39
-d (0.25 + 0.40) = 39
-d (0.65) = 39
-d = \(\frac{39}{0.65}\)
-d = \(\frac{39 × 100}{65}\)
-d = 60
d = -60
Hence, from the above,
We can conclude that the value of d is: -60

b. -9.760 – (-12.81d) = 8.54
Answer:
The given expression is:
-9.760 – (-12.81d) = 8.54
-9.760 + 12.81d = 8.54
Rearrange the like terms in the above equation
So,
12.81d = 8.54 + 9.760
12.81d = 18.3
Divide by 12.81 on both sides
So,
\(\frac{12.81d}{12.81}\) = \(\frac{18.3}{12.81}\)
d = 1.428
Hence, from the above,
We can conclude that the value of d is: 1.428

KEY CONCEPT

In an equation with variable terms on one side, you can combine like terms before using inverse operations and properties of equality to solve the equation.
0.8n + 0.6n = 42
1.4n = 42 → Combine like terms.
\(\frac{1.4 n}{1.4}=\frac{42}{1.4}\)
n = 30

Do You Understand?
Question 1.
Essential Question How do you solve equations that contain like terms?
Answer:
In the equations that contain “Like terms”,
First, arrange the like terms at one side i.e., either the left side or the right side and combine them and then solve the equation for the desired result

Question 2.
Look for Relationships How do you recognize when an equation has like terms?
Answer:
We know that,
“Like terms” are terms that have the same variables raised to the same exponents.
Hence,
When there are the same variables in the given equation, we can call that terms “Like terms” in the given equation

Question 3.
Make Sense and Persevere in the equation 0.755 – \(\frac{5}{8}\)s = 44, how do you combine the like terms?
Answer:
The given equation is:
0.755 – \(\frac{5}{8}\)s = 44
We know that,
“Like terms” are terms that have the same variables raised to the same exponents.
So,
In the given equation,
0.755 and 44 are the like terms
So,
\(\frac{5}{8}\)s = 0.755 + 44
\(\frac{5}{8}\)s = 44.755
Multiply with \(\frac{8}{5}\) on both sides
So,
\(\frac{5}{8}\)s × \(\frac{8}{5}\) = 44.755 × \(\frac{8}{5}\)
s = 71.608
Hence, from the above,
We can conclude that the value of s is: 71.608

Do You Know How?
Question 4.
Henry is following the recipe card to make a cake. He has 95 cups of flour. How many cakes can Henry make?

Answer:
It is given that Henry is following the recipe card to make a cake and he has 95 cups of flour
It is also given that
We need
2\(\frac{2}{3}\) cups of flour for the batter
\(\frac{1}{2}\) cup of flour for the topping
Now,
Let the number of cakes be x
So,
By using the flour for the batter and the topping, Henry can make x cakes
Now,
(2\(\frac{2}{3}\) + \(\frac{1}{2}\))x = 95
We know that,
2\(\frac{2}{3}\) = \(\frac{8}{3}\)
So,
(\(\frac{8}{3}\) + \(\frac{1}{2}\))x = 95
\(\frac{19}{6}\)x = 95
Multiply with \(\frac{6}{19}\) on both sides
So,
\(\frac{19}{6}\)x × \(\frac{6}{19}\) = 95 × \(\frac{6}{19}\)
x = \(\frac{95 × 6}{19}\)
x = 30
Hence, from the above,
We can conclude that the number of cakes made by Henry is: 30

Question 5.
A city has a population of 350,000. The population has decreased by 30% in the past ten years. What was the population of the city ten years ago?
Answer:
It is given that a city has a population of 350,000 and it has decreased by 30% in the past ten years
Now,
Let the population of the city ten years ago be: x
To find the population of the city ten years ago,
We have to find the value of 30% of 350,000 and add it to the 350,000
The reason is it is given that the population i.e., 350,000 decreased in the past ten years. So, the population will be more than 350,000 ten years ago
So,
The population of the city ten years ago = (The population of the city in the past ten years) + (The value of 30% of 350,000)
x = 350,000 + \(\frac{30}{100}\) × 350,000
x = 350,000 + \(\frac{30 × 350,000}{100}\)
x = 350,000 + 105,000
x = 455,000
Hence, from the above,
We can conclude that the population of the city ten years ago is: 455,000

Question 6.
Solve the equation –12.2z – 13.4z = -179.2.
Answer:
The given equation is:
-12.2z – 13.4z = -179.2
From the given equation,
We can observe that 12.2 and 13.4 are the like terms
So,
-z(12.2 + 13.4) = -179.2
z(12.2 + 13.4) = 179.2
z(25.6) = 179.2
Divide by 25.6 into both sides
So,
\(\frac{25.6}{25.6}\)z = \(\frac{179.2}{25.6}\)
z = 7
Hence, from the above,
We can conclude that the value of z is: 7

Practice & Problem Solving

Leveled Practice In 7 and 8, complete the steps to solve for x.
Question 7.
\(\frac{4}{5}\)x – \(\frac{1}{4}\)x = 11

Answer:
The given equation is:
\(\frac{4}{5}\)x – \(\frac{1}{4}\)x = 11
x (\(\frac{4}{5}\) – \(\frac{1}{4}\)) = 11
x (\(\frac{16 – 5}{20}\)) = 11
\(\frac{11}{20}\)x = 11
Multiply with \(\frac{20}{11}\) on both sides
So,
\(\frac{20}{11}\) (\(\frac{11}{20}\)x) = 11 × \(\frac{20}{11}\)
x = \(\frac{11 × 20}{11}\)
x = 20
Hence, from the above,
We can conclude that the value of x is: 20

Question 8.
-0.65x + 0.45x = 5.4

Answer:
The given equation is:
-0.65x + 0.45x = 5.4
So,
x (0.45 – 0.65) = 5.4
x (-0.20) =5.4
Divide by -0.20 into both sides
So,
\(\frac{-0.20}{-0.20}\)x = \(\frac{5.4}{-0.20}\)
x = -27
Hence, from the above,
We can conclude that the avlue of x is: -27

In 9-12, solve for x.
Question 9.
\(\frac{4}{9}\)x + \(\frac{1}{5}\)x = 87
Answer:
The given equation is:
\(\frac{4}{9}\)x + \(\frac{1}{5}\)x = 87
So,
x (\(\frac{4}{9}\) + \(\frac{1}{5}\)) = 87
x (\(\frac{20 + 9}{45}\)) = 87
\(\frac{29}{45}\)x = 87
Multiply with \(\frac{45}{29}\) on both sides
So,
\(\frac{45}{29}\) (\(\frac{29}{45}\)x) = 87 × \(\frac{45}{29}\)
x = \(\frac{87 × 45}{29}\)
x = 135
Hence, from the above,
We can conclude that the value of x is: 135

Question 10.
-3.8x – 5.9x = 223.1
Answer:
The given equation is:
-3.8x – 5.9x = 223.1
So,
-x (3.8 + 5.9) = 223.1
-x (9.7) =223.1
Divide by -9.7 into both sides
So,
\(\frac{-9.7}{-9.7}\)x = \(\frac{223.1}{-9.7}\)
x = -23
Hence, from the above,
We can conclude that the avlue of x is: -23

Question 11.
x + 0.15x = 3.45
Answer:
The givene quation is:
x + 0.15x = 3.45
So,
x (1 + 0.15) = 3.45
x (1.15) = 3.45
Divide be 1.15 into both sides
So,
\(\frac{1.15}{1.15}\)x = \(\frac{3.45}{1.15}\)
x = 3
Hence, from the above,
We can conclude that the value of x is: 3

Question 12.
–\(\frac{3}{5}\)x – \(\frac{7}{10}\) + \(\frac{1}{2}\)x = 56
Answer:
The given equation is:
–\(\frac{3}{5}\)x – \(\frac{7}{10}\) + \(\frac{1}{2}\)x = 56
x (\(\frac{1}{2}\) – \(\frac{3}{5}\)) – \(\frac{7}{10}\) = 56
x (\(\frac{5 – 6}{10}\)) – \(\frac{7}{10}\) = 56
–\(\frac{1}{10}\)x = 56 + \(\frac{7}{10}\)
–\(\frac{1}{10}\)x = \(\frac{560 + 7}{10}\)
Multiply with 10 on both sides
So,
–\(\frac{10}{10}\)x = \(\frac{567 × 10}{10}\)
-x = 567
x = -567
Hence, from the above,
We can conclude that the value of x is: -567

Question 13.
A contractor buys 8.2 square feet of sheet metal. She used 2.1 square feet so far and has $183 worth of sheet metal remaining. Write and solve an equation to find out how many sheets of metal costs per square foot.
Answer:
It is given that a contractor buys 8.2 square feet of sheet metal. She used 2.1 square feet so far and has $183 worth of sheet metal remaining.
So,
The remaining square feet of sheet metal = (Total square feet of sheet metal) – (The total square feet of sheet metal used so far)
The remaining square feet of sheet metal = 8.2 – 2.1
The remaining square feet of sheet metal = 6.1 square feet
Now,
It is given that there is$183 worth of sheet metal remaining
Now,
Let x be the number of sheet metals per square foot
So,
6.1x = $183
Divide by 6.1 into both sides
So,
\(\frac{6.1}{6.1}\)x = \(\frac{$183}{6.1}\)
x = 30
Hence, from the above,
We can conclude that the number of metal sheets per square foot is: 30

Question 14.
Make Sense and Persevere Clint prepares and sells trail mixes at his store. This week, he uses \(\frac{3}{8}\) his supply of raisins to make regular trail mix and \(\frac{1}{4}\) of his supply to make spicy trail mix. If Clint uses 20 pounds of raisins this week, how many pounds of raisins did he have at the beginning of the week?
Answer:
It is given that Clint prepares and sells trail mixes at his store and this week, he uses \(\frac{3}{8}\) his supply of raisins to make regular trail mix and \(\frac{1}{4}\) of his supply to make spicy trail mix.
So,
The total amount of raisins to make trail mix = (The supply of raisins to make regular mix) + (The supply of raisins to make spicy mix)
The total amount of raisins to make trail mix = \(\frac{3}{8}\) + \(\frac{1}{4}\)
The total amount of raisins to make trail mix = \(\frac{5}{8}\)
Now,
Let the number of pounds of raisins at the beginning of the week be x
So,
\(\frac{5}{8}\)x = 20
Multiply with \(\frac{8}{5}\) on both sides
So,
x = 20 × \(\frac{8}{5}\)
x = \(\frac{20 × 8}{5}\)
x = 32 pounds
Hence, from the above,
We can conclude that the number of pounds of raisins at the beginning of the week is: 32 pounds

Question 15.
Make Sense and Persevere A submarine descends to \(\frac{1}{6}\) of its maximum depth. Then it descends another \(\frac{2}{3}\) of its maximum depth. If it is now at 650 feet below sea level, what is its maximum depth?

Answer:
It is given that a submarine descends to \(\frac{1}{6}\) of its maximum depth and then it descends another \(\frac{2}{3}\) of its maximum depth and it is now at 650 feet below sea level
Now,
Let x be the maximum depth
So,
\(\frac{1}{6}\)x + \(\frac{2}{3}\)x = 650
\(\frac{1 + 4}{6}\)x = 650
\(\frac{5}{6}\)x = 650
Multiply with \(\frac{6}{5}\) on both sides
So,
x = 650 × \(\frac{6}{5}\)
x = \(\frac{650 × 6}{5}\)
x = 780 feet
Hence, from the above,
We can conclude that the maximum depth is: 780 feet

Question 16.
Model with Math Write an equation that can be represented by the bar diagram, then solve.

Answer:
The given bar diagram is:

So,
From the bar diagram,
The representation of the equation is:
-1.2y + (-4.2y) = -3.78
-1.2y – 4.2y = -3.78
– (1.2y + 4.2y) = -3.78
1.2y + 4.2y = 3.78
5.4y = 3.78
Divide by 5.4 into both sides
So,
\(\frac{5.4}{5.4}\)y = \(\frac{3.78}{5.4}\)
y = 0.7
Hence, from the above,
We can conclude that the value of y is: 0.7

Question 17.
Higher Order Thinking Solve \(\frac{2}{3}\)h – 156 = 3\(\frac{13}{24}\).
Answer:
The given equation is:
\(\frac{2}{3}\)h – 156 = 3\(\frac{13}{24}\)
We know that,
3\(\frac{13}{24}\) = \(\frac{85}{24}\)
So,
\(\frac{2}{3}\)h – 156 = \(\frac{85}{24}\)
\(\frac{2}{3}\)h = \(\frac{85}{24}\) + 156
0.666h = 3.541 + 156
0.666h = 159.541
Divide by 0.666 into both sides
So,
h = \(\frac{159.541}{0.666}\)
h = 239.552
Hence, from the above,
We can conclude that the value of ‘h’ is: 239.552

Question 18.
Model with Math Nathan bought one notebook and one binder for each of his college classes. The total cost of the notebooks and binders was $27.08. Draw a bar diagram to represent the situation. How many classes is Nathan taking?

Answer:
It is given that Nathan bought one notebook and one binder for each of his college classes. The total cost of the notebooks and binders was $27.08.
Now,
Let the number of notebooks and binders that Nathan bought be x
From the figure,
It is given that
The cost of 1 notebook is: $0.95
The cost of 1 binder is: $5.82
So,
The representation of the cost of total notebooks and binders in the form of the equation is:
$0.95x + $5.82x = $27.08
Hence,
The representation of the above equation in the form of a bar diagram is:

Assessment Practice
Question 19.
Construct Arguments Your friend incorrectly says the solution to the equation –\(\frac{3}{5}\)y – \(\frac{1}{7}\)y = 910 is y = 676. What error did your friend make?
A. Added –\(\frac{1}{7}\) to –\(\frac{3}{5}\)
B. Subtracted \(\frac{1}{7}\) from –\(\frac{3}{5}\)
C. Multiplied 910 by \(\frac{26}{35}\)
D. Multiplied 910 by \(\frac{35}{26}\)
Answer:
The given equation is:
–\(\frac{3}{5}\)y – \(\frac{1}{7}\)y = 910
-y (\(\frac{3}{5}\) + \(\frac{1}{7}\)) = 910
–\(\frac{26}{35}\)y = 910
Multiply with –\(\frac{35}{26}\) on both sides
So,
y = -910 × \(\frac{35}{26}\)
y = -1,225
Hence from the above,
We can conclude that the error your friend makes is:
Multiplied 910 by \(\frac{26}{35}\)

Question 20.
A 132-inch board is cut into two pieces. One piece is three times the length of the other. Find the length of the shorter piece.
PART A
Draw a bar diagram to represent the situation.
Answer:
It is given that a 132-inch board is cut into two pieces and one piece is 3 times the length of the other
Now,
Let the length of 1 piece be x inches
So,
The length of the other piece is: 3x inches
So,
The representation of the given situation in the form of an equation is:
3x + x = 132
Hence,
The representation of the above equation in the form of a bar diagram is:

PART B
Write and solve an equation to find the length of the shorter piece.
Answer:
From part (a),
The equation that represents the given situation is:
3x + x = 132
4x = 132
Divide by 4 into both sides
So,
x = \(\frac{132}{4}\)
x = 33 inches
Hence,from the above,
We can conclude that the length of the shorter piece is: 33 inches

Lesson 2.2 Solve Equations with Variables on Both Sides

Solve & Discuss It!
Jaxson and Bryon collected an equal amount of money during a car wash. They collected cash and checks as shown below. If each check is written for the same amount, x, what is the total amount of money collected by both boys? Explain.

I can… solve equations with variables on both sides of the equal sign.
Answer:
It is given that Jaxson and Bryon collected an equal amount of money during the car wash.
It is also given that they collected cash and checks and each check is written for the same amount x
So,
The amount earned by Jaxson = The amount earned by Bryon
Now,
From the given figure,
The amount earned by Jaxson = The amount earned by cash + The amount earned by checks
= 15 + 14x
The amount earned by Bryon = The amount earned by cash + The amount earned by checks
= 50 + 7x
So,
Now,
15 + 14x = 50 + 7x
Subtract with 7x on both sides
15 + 14x – 7x = 50 + 7x – 7x
15 + 7x = 50
Subtract with 15 on both sides
15 + 7x – 15 = 50 – 15
7x = 35
Divide by 7 on both sides
\(\frac{7}{7}\)x = \(\frac{35}{7}\)
x = 5
So,
The total amount of money collected by both boys = 15 +14x + 50 + 7x
= 21x + 65
= 21 (5) + 65
= 105 + 65
= $170
Hence, from the above,
We can conclude that the total amount earned by both the boys is: $170

Reasoning
How can you use an equation to show that expressions are equal?
Answer:
Combine any like terms on each side of the equation i.e., x-terms with x-terms and constants with constants. Arrange the terms in the same order, usually x-term before constants.
If all of the terms in the two expressions are identical, then the two expressions are equivalent.

Focus on math practices
Model with Math What expressions can you write to represent the amount of money collected by each boy? How can you use these expressions to write an equation?
Answer:
From the given figure,
We can observe that the two boys earned cash and checks
So,
The total amount earned by any boy = The amount earned due to cash + The amount earned due to checks
Now,
The amount earned by Jaxson = The amount earned by cash + The amount earned by checks
= 15 + 14x
The amount earned by Bryon = The amount earned by cash + The amount earned by checks
= 50 + 7x
Now,
It is given that the amount earned by both boys are equal
So,
The amount earned by Jaxson = The amount earned by Bryon
15 + 14x = 50 + 7x
Rearrange the like terms
14x – 7x = 50 – 15
7x = 35
Hence, from the above,
We can conclude that the representation of the amount collected by each boy in the form of the equation is:
7x = 35

Essential Question
How do you use inverse operations to solve equations with variables on both sides?
Answer:
The “Inverse operations” allow us to undo what has been done to the variable
Example:
Solve:
$x+3=\mathrm{8\; From\; the\; above\; equation,\; We\; can\; observe\; that}$$3$ has been added to the variable, $x$.
We know that,
The inverse of addition is subtraction
So,
By subtracting $3$, We can undo the addition.
Now,
After $3$ was added, the result was equal to $8$.
We undo the addition, by subtracting $3$ and see that, the starting amount was $5$

Try It!

Class A was given a sunflower with a height of 8 centimeters that grows at a rate of 3\(\frac{1}{2}\) centimeters per week. Class B was given a sunflower with a height of 10 centimeters that grows at a rate of 3\(\frac{1}{4}\) centimeters per week. After how many weeks are the sunflowers the same height?
Let w= the number of weeks.
____ w + 8 = _____ w + 10
_____ w + 8 = 10
_____ w = _____
w = _____
The sunflowers are the same height after ________ weeks.
Answer:
It is given that
Class A was given a sunflower with a height of 8 centimeters that grows at a rate of 3\(\frac{1}{2}\) centimeters per week and class B was given a sunflower with a height of 10 centimeters that grows at a rate of 3\(\frac{1}{4}\) centimeters per week.
Now,
Let the number of weeks be w
So,
The height of a sunflower of class A after w weeks = 3\(\frac{1}{2}\)w + 8
We know that,
3\(\frac{1}{2}\) = \(\frac{7}{2}\)
So,
The height of a sunflower of class A after w weeks = \(\frac{7}{2}\)w + 8
Now,
The height of a sunflower of class B after w weeks = 3\(\frac{1}{4}\)w + 10
We know that,
3\(\frac{1}{4}\) = \(\frac{13}{4}\)
So,
The height of a sunflower of class A after w weeks = \(\frac{13}{4}\)w + 10
Now,
To make the height of a sunflower from both classes equal,
The height of sunflower of class A after w weeks = The height of sunflower of class B after w weeks
\(\frac{7}{2}\)w + 8 = \(\frac{13}{4}\)w + 10
Rearrange the like terms
\(\frac{7}{2}\)w – \(\frac{13}{4}\)w = 10 – 8
\(\frac{14 – 13}{4}\)w = 2
\(\frac{1}{4}\)w = 2
Multiply with 4 on both sides
\(\frac{4}{4}\)w = 2 (4)
w = 8 weeks
Hence, from the above,
We can conclude that after 8 weeks, the sunflowers of class A and class B are of the same height

Convince Me!
How can you check your work to make sure the value of the variable makes the equation true? Explain.
Answer:
To make a true equation, check your math to make sure that the values on each side of the equals sign are the same. Ensure that the numerical values on both sides of the “=” sign are the same to make a true equation.
Examples:
a) 9 = 9 is a true equation
b) 5 + 4 = 9 is a true equation

Try It!

Solve the equation 96 – 4.5y – 3.2y = 5.6y + 42.80.
Answer:
The given equation is:
96 – 4.5y – 3.2y = 5.6y + 42.80
Now,
Rearrange the like terms at one side i.e., y-terms to one side and the constant terms to other side
So,
-5.6y – 4.5y – 3.2y = 42.80 – 96
-13.3y = -53.2
13.3y = 53.2
Divide by 13.3 into both sides
So,
\(\frac{13.3}{13.3}\)y = \(\frac{53.2}{13.3}\)
y = 4
Hence, from the above,
We can conclude that the value of y is: 4

KEY CONCEPT

When two expressions represent equal quantities, they can be set equal to each other. Then you can use inverse operations and properties of equality to combine like terms and solve for the unknown.

3x + 15 = 4x + 12
3x – 3x + 15 = 4x – 3x + 12
15 = x + 12
15 – 12 = x + 12 – 12
3 = x

Do You Understand?
Question 1.
Essential Question How do you use inverse operations to solve equations with variables on both sides?
Answer:
The “Inverse operations” allow us to undo what has been done to the variable
Example:
Solve:
$x+3=\mathrm{8\; From\; the\; above\; equation,\; We\; can\; observe\; that}$$3$ has been added to the variable, $x$.
We know that,
The inverse of addition is subtraction
So,
By subtracting $3$, We can undo the addition.
Now,
After $3$ was added, the result was equal to $8$.
We undo the addition, by subtracting $3$ and see that, the starting amount was $5$

Question 2.
Reasoning Why are inverse operations and properties of equality important when solving equations? Explain.
Answer:
An “Inverse operation” is two operations that undo each other
Ex: Addition and Subtraction or Multiplication and Division.
You can perform the same inverse operation on each side of an equivalent equation without changing the equality.
This gives us a couple of properties that hold true for all equations.

Question 3.
Model with Math Cynthia earns $680 in commissions and is paid $10.25 per hour. Javier earns $410 in commissions and is paid $12.50 per hour. What will you find if you solve for x in the equation 10.25x + 680 = 12.5x + 410?
Answer:
It is given that
Cynthia earns $680 in commissions and is paid $10.25 per hour. Javier earns $410 in commissions and is paid $12.50 per hour.
It is also given that
The representation of the given situation in the form of the equation is:
10.25x + 680 = 12.5x + 410
From the above equation,
We can observe that
10.25x is the amount paid to Cynthia per hour and x is the number of hours
Hence, from the above,
We can conclude that the variable x represents the “Number of hours”

Do You Know How?
Question 4.
Maria and Liam work in a banquet hall. Maria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 plus a 10% commission on his food sales. What amount of food sales will result in Maria and Liam earning the same amount for the week?
Answer:
It is given that
Maria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 plus a 10% commission on his food sales.
So,
To find the number of food sales that will result in Maria and Liam earning the same amount for the week,
20%x = $625 + 10%x
Where,
x is the number of food sales
So,
\(\frac{20}{100}\)x = $625 + \(\frac{10}{100}\)x
Rearrange the like terms
\(\frac{20 – 10}{100}\)x = $625
\(\frac{10}{100}\)x = $625
\(\frac{1}{10}\)x = $625
Multiply with 10 on both sides
So,
\(\frac{10}{10}\)x = $625 (10)
x = $6,250
Hence, from the above,
We can conclude that the number of food sales that will make the same amount in the week for Maria and Liam is: $6,250

Question 5.
Selma’s class is making care packages to give to victims of a natural disaster. Selma packs one box in 5 minutes and has already packed 12 boxes. Her friend Trudy packs one box in 7 minutes and has already packed 18 boxes. How many more minutes does each need to work in order to have packed the same number of boxes?
Answer:
It is given that
Selma’s class is making care packages to give to victims of a natural disaster. Selma packs one box in 5 minutes and has already packed 12 boxes. Her friend Trudy packs one box in 7 minutes and has already packed 18 boxes.
Now,
Let x be the number of  more minutes that each has to work so that they have the same number of boxes
So,
To find the more minutes each need to work in order to have packed the same number of boxes,
\(\frac{x}{5}\) + 12 = \(\frac{x}{7}\) + 18
Rearrange the like terms
So,
\(\frac{x}{5}\) – \(\frac{x}{7}\) = 18 – 12
\(\frac{7x – 5x}{35}\) = 6
\(\frac{2x}{35}\) = 6
Divide by 35 into both sides
So,
2x = 6 (35)
Divide by 2 into both sides
So,
x = \(\frac{6 (35)}{2}\)
x = 3 (35)
x = 105 minutes
Hence, from the above,
We can conclude that the number of more minutes that each need to work so that the number of boxes becomes equal is: 105 minutes

Question 6.
Solve the equation –\(\frac{2}{5}\)x + 3 = \(\frac{2}{3}\)x + \(\frac{1}{3}\).
Answer:
The given equation is:
–\(\frac{2}{5}\)x + 3 = \(\frac{2}{3}\)x + \(\frac{1}{3}\)
Rearrange the like terms
So,
\(\frac{2}{3}\)x + \(\frac{2}{5}\)x = 3 – \(\frac{1}{3}\)
\(\frac{10 + 6}{15}\)x = \(\frac{9 – 1}{3}\)
\(\frac{16}{15}\)x = \(\frac{8}{3}\)
Multiply with \(\frac{15}{16}\) on both sides
x = \(\frac{8}{3}\) × \(\frac{15}{16}\)
x = \(\frac{8 × 15}{3 × 16}\)
x = \(\frac{5}{2}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{5}{2}\)

Question 7.
Solve the equation -2.6b + 4 = 0.9b – 17.
Answer:
The given equation is:
-2.6b + 4 = 0.9b – 17
Rearrange the like terms
So,
0.9b + 2.6b = 17 + 4
3.5b = 21
Divide by 3.5 into both sides
So,
\(\frac{3.5}{3.5}\)b = \(\frac{21}{3.5}\)
b = 6
Hence, from the above,
We can conclude that the value of b is: 6

Practice & Problem Solving

Leveled Practice In 8 and 9, solve each equation.
Question 8.
6 – 4x = 6x – 8x + 2
6 – 4x = ____ + 2
6 = _____ + 2
____ = _____
_______ = x
Answer:
The given equation is:
6 – 4x = 6x – 8x + 2
So,
6 – 4x = 2 – 2x
Rearrange the like terms
So,
4x – 2x = 6 – 2
2x = 4
Divide by 2 into both sides
So,
\(\frac{2}{2}\)x = \(\frac{4}{2}\)
x = 2
Hence, from the above,
We can conclude that the value of x is: 2

Question 9.

Answer:
The given equation is:
\(\frac{5}{3}\)x + \(\frac{1}{3}\)x = 13\(\frac{1}{3}\) + \(\frac{8}{3}\)x
Rearrange the like terms
So,
\(\frac{5 + 1}{3}\)x – \(\frac{8}{3}\)x = 13\(\frac{1}{3}\)
\(\frac{6 – 8}{3}\)x = 13\(\frac{1}{3}\)
–\(\frac{2}{3}\)x = \(\frac{40}{3}\)
Multiply with 3 on both sides
So,
-2x = 40
divide by -2 into both sides
So,
x = \(\frac{-40}{2}\)
x = -20
Hence, from the above,
We can conclude that the value of x is: -20

Question 10.
Two towns have accumulated different amounts of snow. In Town 1, the snow depth is increasing by 3\(\frac{1}{2}\) inches every hour. In Town 2, the snow depth is increasing by 2\(\frac{1}{4}\) inches every hour. In how many hours will the snowfalls of the towns be equal?

Answer:
It is given that
Two towns have accumulated different amounts of snow. In Town 1, the snow depth is increasing by 3\(\frac{1}{2}\) inches every hour. In Town 2, the snow depth is increasing by 2\(\frac{1}{4}\) inches every hour.
Now,
Let x be the number of hours
So,
To make the snowfalls of the two towns equal,
5 + 3\(\frac{1}{2}\)x = 6 + 2\(\frac{1}{4}\)x
We know that,
3\(\frac{1}{2}\) = \(\frac{7}{2}\)
2\(\frac{1}{4}\) = \(\frac{9}{4}\)
So,
\(\frac{7}{2}\)x – \(\frac{9}{4}\)x = 6 – 5
\(\frac{14 – 9}{4}\)x = 1
\(\frac{5}{4}\)x = 1
Multiply with \(\frac{4}{5}\) on both sides
So,
x = \(\frac{4}{5}\)
x = 0.8 hours
Hence, from the above,
We can conclude that after 0.8 hours, the snowfalls of the two towns will be equal

Question 11.
Solve the equation 5.3g + 9 = 2.3g + 15.
a. Find the value of g.
Answer:
The given equation is:
5.3g + 9 = 2.3g + 15
Rearrange the like terms
So,
5.3g – 2.3g = 15 – 9
3.0g = 6
Divide by 3 into both sides
\(\frac{3}{3}\)g = \(\frac{6}{3}\)
g = 2
Hence, from the above,
We can conclude that the value of g is: 2

b. Explain how you can check that the value · you found for g is correct. If your check does not work, does that mean that your result is incorrect? Explain.
Answer:
From part (a),
We get the value of g : 2
So,
Whether the value of g is correct or not, put it in the given equation
If LHS = RHS,
Then, your check is correct. Otherwise, your check is not correct
Now,
5.3g + 9 = 2.3g + 15
Put, g = 2
So,
5.3 (2) + 9 = 2.3 (2) + 15
10.6 + 9 = 4.6 + 15
19.6 = 19.6
Hence, from the above,
We can conclude that the check is correct

Question 12.
Solve the equation 6 – 6x = 5x – 9x – 2.
Answer:
The given equation is:
6 – 6x = 5x – 9x – 2
So,
6 – 6x = -4x – 2
Rearrange the like terms
So,
-4x + 6x = 6 + 2
2x = 8
Divide by 2 into both sides
So,
\(\frac{2}{2}\)x = \(\frac{8}{2}\)
x = 4
Hence, from the above,
We can conclude that the value of x is: 4

Question 13.
Model with Math The population of one town in Florida is 43,425. About 125 people move out of the town each month. Each month, 200 people on average move into town. A nearby town has a population of 45,000. It has no one moving in and an average of 150 people moving away every month. In about how many months will the population of the towns be equal? Write an equation that represents this situation and solve it.
Answer:
It is given that
The population of one town in Florida is 43,425. About 125 people move out of the town each month. Each month, 200 people on average move into town. A nearby town has a population of 45,000. It has no one moving in and an average of 150 people moving away every month.
Now,
Let the population that are moving in and moving out be x
We know that,
Moving in will be positive and Moving out will be negative
So,
The population of one town in Florida = 43,425 + 200x – 125x
The population of a nearby town = 45,000 – 150x
So,
To find out after how many months, they will be equal,
43,425 + 200x – 125x = 45,000 – 150x
43,425 + 75x = 45,000 – 150x
Rearrange the like terms
So,
150x + 75x = 45,000 – 43,425
225x = 1,575
Divide by 225 into both sides
\(\frac{225}{225}\)x = \(\frac{1,575}{225}\)
x = 7
Hence, from the above,
We can conclude that after 7 months, the population of the towns will be equal

Question 14.
Veronica is choosing between two health clubs After how many months will the total cost for each health club be the same?

Answer:
It is given that Veronica is choosing between two health clubs
Now,
Let x be the number of months so that the cost for the two health clubs will be the same
Now,
The total health cost of Yoga studio A = 22 + 24.50x
The total health cost of Yoga studio B = 47 + 18.25x
So,
To find out after how many months, the total cost for the two health clubs will be the same,
22 + 24.50x = 47 + 18.25x
Rearrange the like terms
So,
47 – 22 = 24.50x – 18.25x
25 = 6.25x
Divide by 25 into both sides
So,
\(\frac{25}{25}\) = \(\frac{6.25}{25}\)x
1 = 0.25x
\(\frac{x}{4}\) = 1
x = 4
Hence, from the above,
We can conclude that after 4 months, the total cost for the two health clubs will be the same

Question 15.
Higher-Order Thinking The price of Stock A at 9 A.M. was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at the rate of $0.14 per hour. If the stocks continue to increase and decrease at the same rates, in how many hours will the prices of the stocks be the same?

Answer:
It is given that
The price of Stock A at 9 A.M. was $12.73. Since then, the price has been increasing at the rate of $0.06 per hour. At noon, the price of Stock B was $13.48. It begins to decrease at the rate of $0.14 per hour.
Now,
Let x be the number of hours
So,
The price of stock A = $12.73 + $0.06x (Since it is increasing)
The price of stock B = $13.48 – $0.14x (Since it is decreasing)
Now,
To find out after how many hours, the prices will be the same,
$12.73 + $0.06x = $13.48 – $0.14x
Rearrange the like terms
So,
$13.48 – $12.73 = $0.14x + $0.06x
$0.75 = $0.2x
Divide by 0.2 into both sides
So,
x = \(\frac{0.75}{0.2}\)
x = 3.75
x = 3.60 + 0.15
x = 4 hours 15 minutes
Hence, from the above,
We can conclude that after 4 hours 15 minutes, the prices of the stocks will be equal

Assessment Practice
Question 16.
In an academic contest, correct answers earn 12 points and incorrect answers lose 5 points. In the final round, School A starts with 165 points and gives the same number of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as School A. The game ends with the two schools tied.
PART A
Which equation models the scoring in the final round and the outcome of the contest?
A. 12x + 5x – 165 = -12x + 65
B. 12x – 5x + 165 = 12x + 65
C. 5x – 12x + 165 = 12x + 65
D. 12x – 5x – 165 = 12x + 65
Answer:
It is given that
In an academic contest, correct answers earn 12 points and incorrect answers lose 5 points. In the final round, School A starts with 165 points and gives the same number of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as School A. The game ends with the two schools tied.
Now,
Let the number of answers be x
We know that,
The points earned for the correct answers will be positive whereas, for the negative answers, they will be negative
So,
For school A,
The number of answers is:
12x – 5x = -165
12x – 5x + 165 = 0
For school B,
The number of answers is:
12x + 0 = -65
12x + 65 = 0
Now,
It is given that the two schools are tied
So,
12x – 5x + 165 = 12x + 65
Hence, from the above,
We can conclude that option B matches the above-given situation

PART B
How many answers did each school get correct in the final round?
Answer:
From part (a),
The equation that models the scoring and outcome of the contest is:
12x – 5x + 165 = 12x + 65
Now,
Rearrange the terms
So,
12x – 12x – 5x = 65 – 165
-5x = -100
5x = 100
Divide by 5 into both sides
So,
\(\frac{5}{5}\)x = \(\frac{100}{5}\)
x = 20
Hence, from the above,
We can conclude that each school gets 20 correct answers in the final round

Lesson 2.3 Solve Multistep Equations

Solve & Discuss It!
A water tank fills through two pipes. Water flows through one pipe at a rate of 25,000 gallons an hour and through the other pipe at 45,000 gallons an hour. Water leaves the system at a rate of 60,000 gallons an hour.

I can… solve multistep equations and pairs of equations using more than one approach.
There are 3 of these tanks, and each tank holds 1 million gallons. Each tank is half full. Water is entering and leaving a tank at the maximum amounts. Determine the number of hours, x, it will take to fill all 3 tanks.
Answer:
It is given that
A water tank fills through two pipes. Water flows through one pipe at a rate of 25,000 gallons an hour and through the other pipe at 45,000 gallons an hour. Water leaves the system at a rate of 60,000 gallons an hour and there are 3 of these tanks, and each tank holds 1 million gallons. Each tank is half full. Water is entering and leaving a tank at the maximum amounts.
Now,
The capacity of each tank = \(\frac{1 million}{2}\) (Since the tank is half-full)
We know that,
1 million = 10 lakhs
So,
The capacity of each tank is: 5 Lakh gallons
So,
The capacity of 3 tanks = 5 Lakh gallons (3)
= 15 Lakh gallons
Now,
The rate of flow of each tank = (The rate of flow of inlet pipes) + (The rate of flow of outlet pipes)
We know that,
The rate of flow for the inlet pipe will be: Positive
The rate of flow for the outlet pipe will be: Negative
So,
The rate of flow of each tank = (45,000 + 25,000) – 60,000
= 70,000 – 60,000
= 10,000 gallons per hour
Since the three pipes are the same, the rate of flow will also be the same
So,
The rate of flow of three tanks = 10,000 (3)
= 30,000 gallons per hour
Now,
It is given that the number of hours is: x
So,
The number of hours took to fill all the three tanks = \(\frac{ The capacity of three tanks } { The rate of flow of the three tanks }\)
x = \(\frac{15,00,000}{30,000}\)
x = 50 hours
Hence, from the above,
We can conclude that the number of hours took to fill the three tanks is: 50 hours

Reasoning
Can you solve the problem in more than one way?
Answer:
Yes, we can solve the problem in more than one way
The first way:
First, calculate the capacity and the rate of flow of each tank and multiply both the quantities with 3 since it is for 3 tanks
So,
We will get the time took to fill the three tanks
The second way:
Calculate the capacity and the rate of the flow of each tank and also find the time taken to fill that tank and multiply the time taken by 3 to get the time taken to fill the three tanks

Focus on math practices
Use Structure What are two different ways to simplify the expression 4(3x + 7x + 5) so that it equals 40x – 20? Explain.
Answer:
The given expression is:
4 (3x + 7x + 5)
A)
The first way:
We know that,
The distributive property is:
a (b + c) = ab + ac
So,
4 (3x + 7x + 5)
= 4 (3x) + 4 (7x) + 4 (5)
= 12x + 28x + 20
= 40x + 20
B)
The second way:
4 (3x + 7x + 5)
First, simplify the expression in the brackets
So,
4 (3x + 7x + 5)
= 4 (10x + 5)
= 4 (10x) + 4(5)
= 40x + 20

Essential Question
How can you use the Distributive Property to solve multistep equations?
Answer:
Let A, B, and C be the three variables
Now,
We know that,
The Distributive Property of multiplication is:
A (B + C) = AB + AC
(A + B) C = AC + BC
(A + C) B = AB + BC

Try It!

Solve the equation 3(x – 5) – 5x = -25 + 6x.
3_____ + 3 ∙ ______ – 5x = – 25 + 6x
_____ – 5x = – 25 + 6x
______ x – 15 = – 25 + 6x
______ – 15 = -25 + _____ x
______ = _____ x ______
x = _____ or ______
Answer:
The given equation is:
3 (x – 5) – 5x = -25 + 6x
3 (x) – 3 (5) – 5x = -25 + 6x
3x – 15 – 5x = -25 + 6x
-15 – 2x = -25 + 6x
Rearrange the like terms
So,
-15 + 25 = 6x + 2x
8x = 10
Divide by 8 into both sides
x = \(\frac{10}{8}\)
x = \(\frac{5}{4}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{5}{4}\)

Convince Me!
Can you add x to -5x on the left side of the equation as the first step? Explain.
Answer:
No, we can’t add x to the -5x because from the given equation,
We are getting
3x – 5x
So,
We have to add 3x and not x to -5x

Try It!

Solve the equation -3(-7 – x) = \(\frac{1}{2}\)(x + 2).
Answer:
The given equation is:
-3 (-7 – x) = \(\frac{1}{2}\) (x + 2)
So,
-3 [-(x + 7)] = \(\frac{1}{2}\) (x + 2)
We know that,
– * – = +
So,
3 (x + 7) = \(\frac{1}{2}\) (x + 2)
Multiply with 2 on both sides
So,
6 (x + 7) = x + 2
6 (x) + 6 (7) = x + 2
6x + 42 = x + 2
Rearrange the like terms
6x – x = 2 – 42
5x = -40
Divide by 5 into both sides
So,
x = \(\frac{-40}{5}\)
x = -8
Hence, from the above,
We can conclude that the value of x is: -8

KEY CONCEPT

When solving multistep equations, sometimes you distribute first and then combine like terms.

Sometimes you combine like terms first and then distribute.

Do You Understand?
Question 1.
Essential Question How can you use the Distributive Property to solve multistep equations?
Answer:
Let A, B, and C be the three variables
Now,
We know that,
The Distributive Property of multiplication is:
A (B + C) = AB + AC
(A + B) C = AC + BC
(A + C) B = AB + BC

Question 2.
Reasoning What is the first step when solving the equation 3(3x – 5x) + 2 = -8?
Answer:
The given equation is:
3 (3x – 5x) + 2 = -8
Use the distributive property of multiplication
So,
3 (3x) – 3(5x) + 2 = -8 ——–> First step when solving the above equation

Question 3.
Use Structure How can you use the order of operations to explain why you cannot combine the variable terms before using the Distributive Property when solving the equation 7(x + 5) – x = 42?
Answer:
The given equation is:
7 (x + 5) – x = 42
To find the order of operations, We have to use the BODMAS rule
So,
From the above equation,
We will first solve the expression present in the brackets, then add, and then subtract
We know that,
We can do any operation only on the like terms
We know that,
The “Like terms” are the terms that have the same exponent
So,
For the above equation,
We can not combine the terms before using the distributive property
Now,
7 (x) + 7 (5) – x = 42
7x + 35 – x = 42
6x + 35 = 42
Subtract with 35 on both sides
6x = 42 – 35
6x = 7
Divide by 6 into both sides
So,
x = \(\frac{7}{6}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{7}{6}\)

Do You Know How?
Question 4.
Solve the equation 3x + 2 = x + 4(x + 2).
Answer:
The given equation is:
3x + 2 = x + 4 (x + 2)
3x + 2 = x + 4 (x) + 4 (2)
3x + 2 = x + 4x + 8
3x + 2 = 5x + 8
Rearrange the like terms
So,
5x – 3x = -8 + 2
2x = -6
Divide by 2 into both sides
So,
x = \(\frac{-6}{2}\)
x = -3
Hence, from the above,
We can conclude that the value of x is: -3

Question 5.
Solve the equation -3(x – 1) + 7x = 27.
Answer:
The given equation is:
-3 (x – 1) + 7x = 27
So,
-3 (x) + 3 (1) + 7x = 27
-3x + 3 + 7x = 27
4x + 3 = 27
Subtract with 3 on both sides
So,
4x = 27 – 3
4x = 24
Divide by 4 into both sides
So,
x = \(\frac{24}{4}\)
x = 6
Hence, from the above,
We can conclude that the value of x is: 6

Question 6.
Solve the equation \(\frac{1}{3}\)(x + 6) = \(\frac{1}{2}\)(x – 3).
Answer:
The given equation is:
\(\frac{1}{3}\)(x + 6) = \(\frac{1}{2}\)(x – 3)
Multiply with 6 on both sides so that we can make the fractions as integers (It is not compulsory to multiply with only 6. You can also multiply with any number that is multiple of both 2 and 3)
So,
\(\frac{6}{3}\) (x + 6) = \(\frac{6}{2}\) (x – 3)
2 (x + 6) = 3 (x – 3)
2 (x) + 2 (6) = 3 (x) – 3 (3)
2x + 12 = 3x – 9
Rearrange the like terms
So,
3x – 2x = 12 + 9
x = 21
Hence,f rom the above,
We can conclude that the value of x is: 21

Question 7.
Solve the equation 0.25(x + 4) – 3 = 28.
Answer:
The given equation is:
0.25 (x + 4) – 3 = 28
Add with 3 on both sides
So,
0.25 (x + 4) = 28 + 3
0.25 (x + 4) = 31
We know that,
0.25 = \(\frac{1}{4}\)
So,
\(\frac{x + 4}{4}\) = 31
Multiply with 4 on both sides
So,
x + 4 = 31 (4)
x + 4 = 124
Subtract with 4 on both sides
So,
x = 124 – 4
x = 120
Hence, from the above,
We can conclude that the value of x is: 120

Practice & Problem Solving

Leveled Practice In 8-10, find the value of x.
Question 8.
Lori bought sunglasses and flip-flops at a half-off sale. If she spent a total of $21 on the two items, what was the original price of the sunglasses?

The original price of the sunglasses was _________.
Answer:
It is given that
Lori bought sunglasses and flip-flops at a half-off sale. If she spent a total of $21 on the two items
Now,
Let x be the price of sunglasses
It is also given that
the price of flipflops is: $24
So,
\(\frac{1}{2}\) (x + $24) = $21
Multiply with 2 on both sides
So,
x + $24 = $21 (2)
x + $24 = $42
Subtract with $24 on both sides
So,
x = $42 – $24
x = $18
Hence, from the above,
We can conclude that the price of sunglasses is: $18

Question 9.
Use the Distributive Property to solve the equation 28 – (3x + 4) = 2(x + 6) + x.
28 – ______ x – _____ = 2x + _____ + x
24 – _____x = ______x + ______
24 – _____x = ______
_____ x = ______
x = ______
Answer:
The given equation is:
28 – (3x + 4) = 2 (x + 6) + x
By using the distributive property,
28 – 3x – 4  = 2 (x) + 2 (6) + x
24 – 3x = 2x + 12 + x
24 – 3x = 3x + 12
Rearrange the like terms
So,
3x + 3x = 24 – 12
6x = 12
Divide by 6 into both sides
So,
x = \(\frac{12}{6}\)
x = 2
Hence, from the above,
We can conclude that the value of x is: 2

Question 10.
Use the Distributive Property to solve the equation 3(x – 6) + 6 = 5x – 6.
x – _____ + 6 = 5x – ______
_____ x – _____ = 5x – _______
_____ x – _____ = _______
______ x = _______
x = ________
Answer:
The given equation is:
3 (x – 6) + 6 = 5x – 6
By using the Distributive property,
3 (x) – 3 (6) + 6 = 5x – 6
3x – 18 + 6 = 5x – 6
3x – 12 = 5x – 6
Rearrange the like terms
So,
5x – 3x = 6 – 12
2x = -6
x = \(\frac{-6}{2}\)
x = -3
Hence, from the above,
We can conclude that the value of x is: -3

Question 11.
What is the solution to -2.5(4x – 4) = -6?
Answer:
The given equation is:
-2.5 (4x – 4) = -6
So,
-2.5 (4x) + 2.5 (4) = -6
-10x + 10 = -6
Subtract with 10 on both sides
So,
-10x = -6 – 10
-10x = -16
10x = 16
Divide by 10 into both sides
So,
x = \(\frac{16}{10}\)
x = 1.6
Hence, from the above,
We can conclude that the solution of the given equation is: 1.6

Question 12.
What is the solution to the equation 3(x + 2) = 2(x + 5)?
Answer:
The given equation is:
3 (x + 2) = 2 (x + 5)
So,
3 (x) + 3 (2) = 2 (x) + 2 (5)
3x + 6 = 2x + 10
Rearrange the like terms
So,
3x – 2x = 10 – 6
x = 4
Hence, from the above,
We can conclude that the solution of the given equation is: 4

Question 13.
Solve the equation \(\frac{1}{6}\)(x – 5) = \(\frac{1}{2}\)(x + 6).
Answer:
The given equation is:
\(\frac{1}{6}\)(x – 5) = \(\frac{1}{2}\)(x + 6)
Multiply with 6 on both sides
So,
x – 5 = 3 (x + 6)
x – 5 = 3 (x) + 3 (6)
x – 5 = 3x + 18
Rearrange the like terms
So,
x – 3x = 18 + 5
-2x = 23
Divide by -2 into both sides
So,
x = –\(\frac{23}{2}\)
Hence, from the above,
We can conclude that the value of x for the given equation is: –\(\frac{23}{2}\)

Question 14.
Solve the equation 0.6(x + 2) = 0.55(2x + 3).
Answer:
The given equation is:
0.6 (x + 2) = 0.55 (2x + 3)
So,
0.6 (x) + 0.6 (2) = 0.55 (2x) + 0.55 (3)
0.6x + 1.2 = 1.10x + 1.65
Rearrange the like terms
So,
1.10x – 0.6x = 1.2 – 1.65
0.5x = -0.45
Divide by 0.5 into both sides
So,
x = \(\frac{-0.45}{0.5}\)
x = -0.9
Hence, from the above,
We can conclud ethat the value of x is: -0.9

Question 15.
Solve the equation 4x – 2(x – 2) = -9 + 5x – 8.
Answer:
The given equation is:
4x – 2 (x – 2) = -9 + 5x – 8
So,
4x – 2 (x) + 2 (2) = -9 + 5x – 8
4x – 2x + 4 = 5x – 17
2x + 4 = 5x – 17
Rearrange the like terms
So,
5x – 2x = 17 + 4
3x = 21
Divide by 3 into both sides
So,
x = \(\frac{21}{3}\)
x = 7
Hence, from the above,
We can conclude that the value of x is: 7

Question 16.
Use the Distributive Property to solve the equation 2(m + 2) = 22. Describe what it means to distribute the 2 to each term inside the parentheses.
Answer:
The given equation is:
2 (m + 2) = 22
We know that,
By using the distributive property of multiplication,
A (B + c) = AB + AC
So,
2 (m) + 2 (2) = 22
2m + 4 = 22
2m = 22 – 4
2m = 18
m = \(\frac{18}{2}\)
m = 9
Hence, from the above,
We can conclude that the value of m is: 9

Question 17.
What is Peter’s number?

Answer:
Let peter’s number be x
So,
According to the given statement,
-3 (x – 12) = -54
3 (x – 12) = 54
3x – 3 (12) = 54
3x – 36 = 54
3x = 36 + 54
3x = 90
x = \(\frac{90}{3}\)
x = 30
Hence, from the above,
We acn conclude that peter’s number is: 30

Question 18.
Higher Order Thinking Use the Distributive Property to solve the equation \(\frac{4x}{5}\) – x = \(\frac{x}{10}\) – \(\frac{9}{2}\)
Answer:
The given equation is:
\(\frac{4x}{5}\) – x = \(\frac{x}{10}\) – \(\frac{9}{2}\)
Rearrange the like terms
So,
\(\frac{4x}{5}\) – x – \(\frac{x}{10}\) = –\(\frac{9}{2}\)
\(\frac{7x}{10}\) – x = –\(\frac{9}{2}\)
–\(\frac{3x}{10}\) = –\(\frac{9}{2}\)
Multiplywith \(\frac{10}{3}\) on both sides
x = \(\frac{9 × 10}{2 × 3}\)
x = 15
Hence, from the above,
We can conclude that the value of x is: 15

Assessment Practice
Question 19.
How many solutions does the equation -2(x + 4) = -2(x + 4) – 6 have?
Answer:
The given equation is:
-2 (x + 4) = -2 (x + 4) – 6
So,
-2 (x) – 2 (4) = -2 (x) – 2 (4) – 6
-2x – 8 = -2x – 8 – 6
Rearrange the like terms
So,
-2x + 2x – 8 + 8 = -6
0 = -6
Hence, from the above,
We can conclude that there are no solutions for the given equation

Question 20.
Solve the equation 3(x + 4) = 2x + 4x – 6 for x.
Answer:
The given equation is:
3 (x + 4) = 2x + 4x – 6
So,
3 (x) + 3 (4) = 6x – 6
3x + 12 = 6x – 6
Rearrange the like terms
So,
6x – 3x = 12 + 6
3x = 18
x = \(\frac{18}{3}\)
x = 6
Hence, from the above,
We can conclude that the solution of the given equation is: 6

Lesson 2.4 Equations with No Solutions or Infinitely Many Solutions

Explore It!
The Great Karlo called twins Jasmine and James onto the stage. Jasmine, multiply your age by 3 and add 6. Then multiply this sum by 2. James, multiply your age by 2 and add 4. Then multiply this sum by 3. I predict you will both get the same number!

I can… determine the number of solutions an equation has.

A. Write expressions to represent Great Karlo’s instructions to each twin.
Answer:
It is given that
The Great Karlo called the twins Jasmine and James onto the stage. Jasmine, multiply your age by 3 and add 6. Then multiply this sum by 2. James, multiply your age by 2 and add 4. Then multiply this sum by 3.
Now,
Great Karlo’s instructions to Jasmine:
Let the age of Jasmine be x
Step 1:
Multiply your age by 3 and add 6
3x + 6
Step 2:
Multiply step 1 with 2
2 (3x +6)
So,
The expression representing the age of Jasmine is: 2 (3x + 6)
Great Karlo’s instructions to James
Let the age of James be x
Step 1:
Multiply your age by 2 and add 4
2x + 4
Step 2:
Multiply step 1 with 3
3 (2x +4)
So,
The expression representing the age of James is: 3 (2x + 4)
Hence, from the above,
We can conclude that the expressions that represent the Great Karlo’s instruction to each twin are:
For Jasmine —–> 2 (3x + 6)
For James ——-> 3 (2x + 4)

B. Choose 4 whole numbers for the twins’ age and test each expression. Make a table to show the numbers you tried and the results.
Answer:
It is given that the great Karlo predicted that the twins will get the same number
So,
2 (3x + 6) = 3 (2x + 4)
2 (3x) + 2 (6) = 3 (2x) + 3 (4)
6x + 12 = 6x + 12
Hence,
The table to show the numbers tried for Jasmine’s and James ages and the results are:

C. What do you notice about your results?
Answer:
From the table that is present in part (b),
We can observe that the ages of Jasmine and James are the same

Focus on math practices
Make Sense and Persevere Choose three more values and use them to evaluate each expression. What do you notice? Do you think this is true for all values? Explain.
Answer:
The table that represents three more values of Jasmine’s and James’ ages and its results are:

Hence, from the above table,
We can observe that the ages of Jasmine and James are the same
Hence, from the above,
We can conclude that for any type of the whole number, the ages of Jasmine and James are the same

Essential Question
Will a one-variable equation always have only one solution?
Answer:
Every linear equation that is a conditional equation has one solution. However, not every linear equation in one variable has a single solution. There are two other cases: no solution and the solution set of all real numbers

Try It!

How many solutions does the equation
3x + 15 = 2x + 10 + x + 5 have?
The equation has ______ solutions.
3x + 15 = 2x + 10 + x + 5
3x + 15 = _____ x + ______
3x – _____ + 15 = 3x – _____ + 15
______ = _______
Answer:
The given equation is:
3x + 15 = 2x + 10 + x + 5
So,
3x + 15 = 3x + 15
Subtract with 3x on both sides
So,
15 = 15
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Convince Me!
If the value of x is negative, would the equation still be true? Explain.
Answer:
For the given equation,
3x + 15 = 2x + 10 + x + 5,
The solutions are infinite i..e, for any value of x, the given equation will be true i.e., for both positive and negative values of x, the equation will be true
Hence, from the above,
We can conclude that the given equation would still be true even if the value of x is negative

Try It!

How many solutions does the equation 4x + 8 = 0.1x + 3 + 3.9x have? Explain.
Answer:
The given equation is:
4x + 8 = 0.1x + 3 + 3.9x
So,
4x + 8 = 4x + 3
Subtract with 4x on both sides
So,
8 = 3
Hence, from the above,
We can conclude that the given equation has no solutions

Try It!

Determine the number of solutions each equation has without solving. Explain your reasoning.
a. 3x + 1.5 = 2.5x + 4.7
Answer:
The give equation is:
3x + 1.5 = 2.5x + 4.7
Rearrange the like terms
So,
3x – 2.5x = 4.7 – 1.5
0.5x = 3.2
Divide by 0.5 into both sides
So,
x = \(\frac{3.2}{0.5}\)
x = 6.4
Hence, from the above,
We can conclude that the given equation ahs only 1 solution

b. 3(x + 2) = 3x – 6
Answer:
The given equation is:
3 (x + 2) = 3x – 6
So,
3 (x) + 3 (2) = 3x – 6
3x + 6 = 3x – 6
Subtract with 3x on both sides
So,
6 = -6
Hence, from the above,
We can conclude that the given equation has no solutions

c. 9x – 4 = 5x – 4 + 4x
Answer:
The given equation is:
9x – 4 = 5x – 4 + 4x
So,
9x – 4 = 9x – 4
Subtract with 9x on both sides
So,
-4 = -4
4 = 4
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

KEY CONCEPT

A one-variable equation has infinitely many solutions when solving results in a true statement, such as 2 = 2.
A one-variable equation has one solution when solving results in one value for the variable, such as x = 2.
A one-variable equation has no solution when solving results in an untrue statement, such as 2 = 3.

Do You Understand?
Question 1.
Essential Question Will a one-variable equation always have only one solution?
Answer:
Every linear equation that is a conditional equation has one solution. However, not every linear equation in one variable has a single solution. There are two other cases: no solution and the solution set of all real numbers

Question 2.
Use Structure Kaylee writes the equation 6x + 12 = 2(3x + 6). Can you find the number of solutions this equation has without solving for x? Explain.
Answer:
The given equation is:
6x + 12 = 2 (3x + 6)
So,
6x + 2 = 2 (3x) + 2 (6)
6x + 12 = 6x + 12
Subtract with 12 on both sides
So,
12 = 12
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 3.
Construct Arguments The height of an experimental plant after x days can be represented by the formula 3(4x + 2). The height of a second plant can be represented by the formula 6(2x + 2). Is it possible that the two plants will ever be the same height? Explain.
Answer:
It is given that
The height of an experimental plant after x days can be represented by the formula 3(4x + 2). The height of a second plant can be represented by the formula 6(2x + 2)
So,
Now,
To find out whether the two plants will ever be the same height or not,
3 (4x + 2) = 6 (2x + 2)
So,
3 (4x) + 3 (2) = 6 (2x) + 6 (2)
12x + 6 = 12x + 12
Subtract with 12x on both sides
So,
6 = 12
So,
The given equation has no solution
Hence, from the above,
We can conclude that it is not possible the two plants will ever be the same height

Do You Know How?
Question 4.
How many solutions does the equation 3(2.4x + 4) = 4.1x + 7 + 3.1x have? Explain.
Answer:
The given equation is:
3 (2.4x + 4) = 4.1x + 7 + 3.1x
So,
3 (2.4x) + 3 (4) = 7.2x + 7
7.2x + 12 = 7.2x + 7
Subtract with 7.2x on both sides
So,
12 = 7
Hence, from the above,
We can conclude that the given equation has no solutions

Question 5.
How many solutions does the equation 7x + 3x – 8 = 2(5x – 4) have? Explain.
Answer:
The given equation is:
7x + 3x – 8 = 2 (5x – 4)
So,
10x – 8 = 2 (5x) – 2 (4)
10x – 8 = 10x – 8
Subtract with 10x on both sides
So,
-8 = -8
8 = 8
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 6.
Todd and Agnes are making desserts. Todd buys peaches and a carton of vanilla yogurt. Agnes buys apples and a jar of honey. They bought the same number of pieces of fruit. Is there a situation in which they pay the same amount for their purchases? Explain.

Answer:
It is given that
Todd and Agnes are making desserts. Todd buys peaches and a carton of vanilla yogurt. Agnes buys apples and a jar of honey. They bought the same number of pieces of fruit.
Now,
Let the number of pieces of fruit be x
So,
The amount purchased by Todd = $1.25x + $4
The amount purchased by Agnes = $1x + $6
Now,
To find whether they pay the same amount for purchase or not,
$1.25x + $4 = $1x + $6
Rearrange the like terms
So,
$1.25x – $1x = $6 – $4
$0.25x = $2
Divide by 0.25 into both sides
So,
x = \(\frac{2}{0.25}\)
x = 8
Hence, from the above,
We can conclude that if there are 8 fruits, then Todd and Agnes will pay the same amount for purchase

Practice & Problem Solving

Leveled Practice In 7 and 8, complete the equations to find the number of solutions.
Question 7.
Classify the equation 33x + 99 = 33x – 99 as having one solution, no solution, or infinitely many solutions.
33x + 99 = 33x – 99
33x – ______ + 99 = 33x – _____ – 99
99 ______ – 99
Since 99 is _______ equal to -99, the equation has _______ solution(s).
Answer:
The given equation is:
33x + 99 = 33x – 99
Subtract with 33x on both sides
So,
33x – 33x + 99 = 33x – 33x – 99
99 = -99
We know that,
99 ≠ -99
Hence, from the above,
We can conclude that there are no solutions for the given equation

Question 8.
Solve 4(4x + 3) = 19x + 9 – 3x + 3. Does the equation have one solution, no solution, or infinitely many solutions?
4(4x + 3) = 19x + 9 – 3x + 3
4 • ______ + 4 • ______ = 19x + 9 – 3x + 3
16x + 12 = _______ + _______
16x – ______ + 12 = 16x ______ + 12
12 _______ 12
Since 12 is ________ equal to 12, the equation has ________ solution(s).
Answer:
The given equation is:
4 (4x + 3) = 19x + 9 – 3x + 3
So,
4 (4x) + 4 (3) = 16x + 12
16x + 12 = 16x + 12
Subtract with 16x on both sides
So,
16x – 16x + 12 = 16x – 16x + 12
12 = 12
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 9.
Generalize What does it mean if an equation is equivalent to 0 = 0? Explain.
Answer:
If an equation is equivalent to 0 = 0, then
The equation is true for all the values of x
Hence,
That equation has infinitely many solutions

Question 10.
Solve 4x + x + 4 = 8x – 3x + 4. Does the equation have one solution, no solution, or infinitely many solutions? If one solution, write the solution. Explain.
Answer:
The given equation is:
4x + x + 4 = 8x – 3x + 4
So,
5x + 4 = 5x + 4
Subtract with x on both sides
So,
5x – 5x + 4 = 5x – 5x + 4
4 = 4
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 11.
Reasoning Two rival dry cleaners both advertise their prices. Let x equal the number of items dry cleaned. Store A’s prices are represented by the expression 15x – 2. Store B’s prices are represented by the expression 3(5x + 7). When do the two stores charge the same rate? Explain.
Answer:
It  is given that
Two rival dry cleaners both advertise their prices. Let x equal the number of items dry cleaned. Store A’s prices are represented by the expression 15x – 2. Store B’s prices are represented by the expression 3(5x + 7)
So,
To find when the two stores charge the same rate,
15x – 2 = 3 (5x + 7)
So,
15x – 2 = 3 (5x) + 3 (7)
15x – 2 = 15x – 21
Subtract with 15x on both sides
So,
15x – 15x – 2 = 15x – 15x – 21
-2 = -21
2 = 21
So,
The equation has no solution
Hence, from the above,
We can conclude that the two stores will never charge the same rate

Question 12.
Reasoning How is solving an equation with no solution similar to solving an equation that has an infinite number of solutions?
Answer:
No solution would mean that there is no answer to the equation. It is impossible for the equation to be true no matter what value we assign to the variable. Infinite solutions would mean that any value for the variable would make the equation true.

Question 13.
Solve 0.9x + 5.1x – 7 = 2(2.5x – 3). How many solutions does the equation have?
Answer:
The given equation is:
0.9x + 5.1x – 7 = 2 (2.5x – 3)
So,
6.0x – 7 = 2 (2.5x) – 2 (3)
6x – 7 = 5x – 6
Rearrange the like terms
So,
6x – 5x = 7 – 6
x = 1
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 14.
Critique Reasoning Your friend solved the equation 4x + 12x – 6 = 4(4x + 7) and got x = 34.
What error did your friend make? What is the correct solution?
4x + 12x – 6 = 4 (4x + 7).
16x – 6 = 16x + 28
16x – 16x – 6 = 16x – 16x + 28
x – 6= 28
x – 6 + 6 = 28 + 6
x = 34
Answer:
The given equation is:
4x + 12x – 6 = 4 (4x + 7)
So,
16x – 6 = 4 (4x) + 4 (7)
16x – 6 = 16x + 28
Subtract with 16x on both sides
So,
16x – 16x – 6 = 16x – 16x + 28
-6 = 28
So,
From the above,
We can observe that after subtracting the given equation with 16x, there are no x terms.
So,
We can’t get the value of x but your friend takes variable x after subtracting 16x from the given equation even though there is no possibility for the x-term
Hence, from the above,
We can conclude that the correct solution for the given equation is: No solutions for the given equation

Question 15.
Solve 49x + 9 = 49x + 83.
a. Does the equation have one solution, no solution, or infinitely many solutions?
Answer:
The given equation is:
49x + 9 = 49x + 83
Subtract with 49x on both sides
So,
49x – 49x + 9 = 49x – 49x + 83
9 = 83
Hence, from the above,
We can conclude that the given equation has no solutions

b. Write two equations in one variable that have the same number of solutions as this equation.
Answer:
The two equations in one variable that have the same number of solutions as the equation that is present in part (a) are:
A) 10x + 8 = 10x – 25
B) 5 (3x + 10) = 15x + 40

Question 16.
Classify the equation 6(x + 2) = 5(x + 7) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
6 (x + 2) = 5 (x + 7)
So,
6 (x) + 6 (2) = 5 (x) + 5 (7)
6x + 12 = 5x + 35
Rearrange the like terms
So,
6x – 5x = 35 – 12
x = 23
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 17.
Solve 6x + 14x + 5 = 5(4x + 1). Write a word problem that this equation, or any of its equivalent forms, represents.
Answer:
The given equation is:
6x + 14x + 5 = 5 (4x + 1)
So,
20x + 5 = 5 (4x) + 5 (1)
20x + 5 = 20x + 5
Subtract with 20x on both sides
So,
20x – 20x + 5 = 20x – 20x + 5
5 = 5
Hence, from the above,
We can conclude that the given equation is true for any value of x i..e, the given equation has infinitely many solutions

Question 18.
Classify the equation 170x – 1,000 = 30(5x – 30) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
170x – 1,000 = 30 (5x – 30)
So,
170x – 1,000 = 30 (5x) – 30 (30)
170x – 1,000 = 150x – 900
Rearrange the like terms
So,
170x – 150x = 1,000 – 900
20x = 100
x= \(\frac{100}{20}\)
x = 5
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 19.
Higher Order Thinking Write one equation that has one solution, one equation that has no solution, and one equation that has infinitely many solutions.
Answer:
The example representation of the equation that has one solution is:
20x + 5 = 15x – 4
The example representation of the equation that has no solutions is:
3 (6x – 2) = 9 (2x – 4)
The example representation of the equation that has infinitely many solutions is:
4 (2x – 6) = 8 (x – 3)

Question 20.
Solve 4(4x – 2) + 1 = 16x – 7.
Answer:
The given equation is:
4 (4x – 2) + 1 = 16x – 7
So,
4 (4x) – 4 (2) + 1 = 16x – 7
16x – 8 + 1 = 16x – 7
16x – 7 = 16x – 7
Subtract with 16x on both sides
So,
16x – 16x – 7 = 16x – 16x – 7
-7 = -7
7 = 7
Hence, from the above,
We can conclude that the given equation is true for all the values of x i..e, the given equation has infinitely many solutions

Question 21.
Solve 6x + 26x – 10 = 8(4x + 10).
Answer:
The given equation is:
6x + 26x – 10 = 8 (4x + 10)
So,
32x – 10 = 8 (4x) + 8 (10)
32x – 10 = 32x + 80
Subtract with 32x on both sides
So,
32x – 32x – 10 = 32x – 32x + 80
-10 = 80
Hence, from the above,
We can conclude that the given equation has no solution

Question 22.
Classify the equation 64x – 16 = 16(4x – 1) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
64x – 16 = 16 (4x – 1)
So,
64x – 16 = 16 (4x) – 16 (1)
64x – 16 = 64x – 16
Subtract with 64x on both sides
So,
64x – 64x – 16 = 64x – 64x – 16
– 16 = -16
16 = 16
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

Question 23.
Classify the equation 5(2x + 3) = 3(3x + 12) as having one solution, no solution, or infinitely many solutions.
Answer:
The given equation is:
5 (2x + 3) = 3 (3x + 12)
So,
5 (2x) + 5 (3) = 3 (3x) + 3 (12)
10x + 15 = 9x + 36
Rearrange the like terms
So,
10x – 9x = 36 – 15
x = 21
Hence, from the above,
We can conclude that the given equation has only 1 solution

Assessment Practice
Question 24.
Which of the following best describes the solution to the equation 4(2x + 3) = 16x + 12 – 8x?
A. The equation has one solution.
B. The equation has infinitely many solutions.
C. The equation has no solution.
D. The equation has two solutions.
Answer:
The given equation is:
4 (2x + 3) = 16x + 12 – 8x
So,
4 (2x) + 4 (3) = 4x + 12
8x + 12 = 4x + 12
Rearrange the like terms
So,
8x – 4x = 12 – 12
4x = 0
x = 0
So,
The given equation has only 1 solution
Hence, from the above,
We can conclude that option A matches with the solution of the given equation

Question 25.
Which of the following statements are true about the equation 10x + 45x – 13 = 11(5x + 6)? Select all that apply.
☐ The operations that can be used to solve the equation are addition and multiplication.
☐ The operations that can be used to solve the equation are multiplication and division.
☐ The equation has infinitely many solutions.
☐ The equation has a solution of x = 53.
☐ The equation has no solution.
Answer:
Let the given options be named: A, B, C, D, and E
Now,
The given equation is:
10x + 45x – 13 = 11 (5x + 6)
So,
55x – 13 = 11 (5x) + 11 (6)
55x – 13 = 55x + 66
Subtract with 55x on both sides
So,
55x – 55x – 13 = 55x – 55x + 66
-13 = 26
Hence, from the above,
We can conclude that option A and option E matches with the situation for the given equation

Topic 2 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you determine the number of solutions for an equation? Lesson 2-4
Answer:
A one-variable equation has infinitely many solutions when solving results in a true statement, such as 2 = 2.
A one-variable equation has one solution when solving results in one value for the variable, such as x = 2.
A one-variable equation has no solution when solving results in an untrue statement, such as 2 = 3.

Question 2.
Solve the equation –\(\frac{2}{3}\)d – \(\frac{1}{4}\)d = -22 for d. Lesson 2-1
Answer:
The given equation is:
–\(\frac{2}{3}\)d – \(\frac{1}{4}\)d = -22
So,
\(\frac{8 + 3}{12}\)d = 22
\(\frac{11}{12}\)d = 22
Multiply with \(\frac{12}{11}\) on both sides
So,
d = 22 × \(\frac{12}{11}\)
d = \(\frac{22 × 12}{11}\)
d = 24
Hence, from the above,
We can conclude that the value of d is: 24

Question 3.
Edy has $450 in her savings account. She deposits $40 each month. Juan has $975 in his checking account. He writes a check for $45.45 each month for his cell phone bill. He also writes a check for $19.55 each month for his water bill. After how many months will Edy and Juan have the same amount of money in their accounts? Lesson 2-2
Answer:
It is given that
Edy has $450 in her savings account. She deposits $40 each month. Juan has $975 in his checking account. He writes a check for $45.45 each month for his cell phone bill. He also writes a check for $19.55 each month for his water bill.
Now,
Let the number of months be x
So,
The amount of money in the account of Edy = $450 + $40x
The amount of money in the account of Juan = $975 – $45.45x – $19.55x
Now,
To find after how many months they will have the same amount of money in their accounts,
$450 + $40x = $975 – $45.45x – $19.55x
$450 + $40x = $975 – $65x
Rearrange the like terms
So,
$65x + $40x = $975 – $450
$105x = $525
Divide by 105 into both sides
So,
x = \(\frac{$25}{105}\)
x = 5 months
Hence, from the above,
We can conclude that after 5 months, Edy and Jian will have the same amount of money in their accounts

Question 4.
Which equation has infinitely many solutions? Lesson 2-4
A. \(\frac{3}{4}\)x + x – 5 = 10 + 2x
Answer:
The given equation is:
\(\frac{3}{4}\)x + x – 5 = 10 + 2x
\(\frac{3 + 4}{4}\)x – 5 = 10 + 2x
\(\frac{7}{4}\)x – 5 = 10 + 2x
Rearrange the like terms
So,
\(\frac{7}{4}\)x – 2x = 10 + 5
–\(\frac{1}{4}\)x = 15
Multiply with -4 on both sides
So,
x = -60
Hence, from the above,
We acn conclude that the given equation has only 1 solution

B. 3x – 2.7 = 2x + 2.7 + x
Answer:
The given equation is:
3x – 2.7 = 2x + x + 2.7
3x – 2.7 = 3x + 2.7
Subtract with 3x on both sides
So,
-2.7 = 2.7
Hence, from the above,
We can conclude that the given equation has no solutions

C. 9x + 4.5 – 2x = 2.3 +7x + 2.2
Answer:
The given equation is:
9x + 4.5 – 2x = 2.3 + 7x + 2.2
7x + 4.5 = 7x + 4.5
Subtract with 7x on both sides
So,
4.5 = 4.5
Hence, from the above,
We can conclude that the given equation has infinitely many solutions

D. \(\frac{1}{5}\) x – 7 = \(\frac{3}{4}\) + 2x – 25\(\frac{3}{4}\)
Answer:
The given equation is:
\(\frac{1}{5}\) x – 7 = \(\frac{3}{4}\) + 2x – 25\(\frac{3}{4}\)
We know that,
25\(\frac{3}{4}\) = \(\frac{103}{4}\)
So,
\(\frac{1}{5}\) x – 7 = \(\frac{3}{4}\) + 2x – \(\frac{103}{4}\)
Rearrange the like terms
So,
\(\frac{1}{5}\)x – 2x = 7 – \(\frac{103}{4}\)
–\(\frac{9}{5}\)x = –\(\frac{75}{4}\)
Multiply with –\(\frac{5}{9}\) on both sides
So,
x = \(\frac{75 × 5}{4 × 9}\)
x = \(\frac{125}{4}\)
Hence, from the above,
We can conclude that the given equation has only 1 solution

Question 5.
Solve the equation -4(x – 1) + 6x = 2(17 – x) for x. Lesson 2.3
Answer:
The given equation is:
-4 (x – 1) + 6x = 2 (17 – x)
So,
-4 (x) + 4 (1) + 6x = 2 (17) – 2 (x)
-4x + 4 + 6x = 34 – 2x
2x + 4 = 34 – 2x
Rearrange the like terms
So,
2x + 2x = 34 – 4
4x = 30
Divide by 4 on both sides
So,
x = \(\frac{30}{4}\)
x = \(\frac{15}{2}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{15}{2}\)

Question 6.
Hakeem subtracted 8 from a number, then multiplied the difference by \(\frac{4}{5}\). The result was 20. Write and solve an equation to find the number, x. Lesson 2-3
Answer:
It is given that
Hakeem subtracted 8 from a number, then multiplied the difference by \(\frac{4}{5}\). The result was 20.
Now,
Let the number be x
So,
According to Hakeem,
The expression that represents the given situation is:
\(\frac{4}{5}\) (x – 8) = 20
Multiply with \(\frac{5}{4}\) on both sides
So,
x – 8 = \(\frac{5 × 20}{4}\)
x – 8 = 25
Add with 8 on both sides
So,
x = 25 + 8
x = 34
Hence, from the baove,
We can conclude that Hakeem’s number is: 34

Topic 2 MID-TOPIC PERFORMANCE TASK

Hector is competing in a 42-mile bicycle race. He has already completed 18 miles of the race and is traveling at a constant speed of 12 miles per hour when Wanda starts the race. Wanda is traveling at a constant speed of 16 miles per hour.

PART A
Write and solve an equation to find when Wanda will catch up to Hector.
Answer:
It is given that
Hector is competing in a 42-mile bicycle race. He has already completed 18 miles of the race and is traveling at a constant speed of 12 miles per hour when Wanda starts the race. Wanda is traveling at a constant speed of 16 miles per hour.
Now,
Let the time be x
We know that,
Speed = \(\frac{Distance} {Time}\)
So,
Time = \(\frac{Distance}{Speed}\)
Now,
Time taken by Hector to complete a bicycle race = \(\frac{42 – 18}{12}\)
x = \(\frac{24}{12}\)
x = 2 hours
Now,
Time taken by Wanda to complete the bicycle race = \(\frac{The total distance of race}{The speed traveled by Wanda}\)
x = \(\frac{42}{16}\)
x = \(\frac{21}{8}\) hours
x = 2.625 hours
Now,
The time that took Wanda to catch up to Hector = The time taken by Wanda to complete the race – The tie taken by Hector to complete the race
= 2.625 – 2
= 0.625
= 0.625 (60 minutes)
= 37.5 minutes
Hence, from the above,
We can conclude that Wanda will catch up to Hector after 37.5 minutes of Hector completing the race

PART B
Will Wanda catch up to Hector before the race is complete? Explain.
Answer:
From part (a),
The time taken by Hector to complete the race is: 2 hours
The time taken by Wanda to complete the race is: 2.625 hours
So,
From the above times,
We can observe that the race is completed at 2 hours
Hence, from the above,
We can conclude that Wanda can’t catch up to Hector before the race is complete

PART C
At what constant speed could Wanda travel to catch up with Hector at the finish line? Explain.
Answer:
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
The speed at which Wanda travel to catch up to Hector = \(\frac{The distance of the race}{The time taken by Hector to complete the race}\)
= \(\frac{42}{2}\)
= 21 miles per hour
Hence, from the above,
We can conclude that at 21 miles per hour speed, Wanda could catch up to Hector

3-Act Mathematical Modeling: Powering Down

3-ACT MATH

АСТ 1

Question 1.
After watching the video, what is the first question that comes to mind?
Answer:
After watching the video,
The first question that comes to mind is:
what will be the battery percentage you should have to complete your work?

Question 2.
Write the Main Question you will answer.

Answer:
The main question you will answer is:
what will be the battery percentage you should have to complete your work?

Question 3.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.
Answer:
The answer to the main question is: 100%
Reason for the prediction:
We don’t know how much work has left. So, it is better to have a battery percentage of 100%

Question 4.
On the number line below, write a time that is too early to be the answer. Write a time that is too late.

Answer:
The time that is too early to be the answer for the above problem is: 5 minutes
The time that is too late to be the answer for the above problem is: Greater than the time that battery percentage is 100%

Question 5.
Plot your prediction on the same number line.

ACT 2

Question 6.
What information in this situation would be helpful to know? How would you use that information?

Answer:
The information in this situation that would be helpful to know is:
A) The time is taken for battery percentage to be full

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math
Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it earlier or later than your prediction? Explain why.

Answer:

ACT 3

Question 10.
Write the answer you saw in the video.
Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?

Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

Act 3

Reflect

Question 13.
Model with Math
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 14.
Look for Relationships What pattern did you notice in the situation? How did you use that pattern?
Answer:

SEQUEL

Question 15.
Be Precise After 35 minutes, he started charging his phone. 21 minutes later, the battery is at 23%. Explain how you would determine when the phone will be charged to 100%.
Answer:

Lesson 2.5 Compare Proportional Relationships

Solve & Discuss It!

Mei Li is going apple picking. She is choosing between two places. The cost of a crate of apples at each place is shown.
Where should Mei Li go to pick her apples? Explain.

Answer:
It is given that Mei Li is going apple picking. She is choosing between two places.
So,
In Annie’s Apple Orchard,
The cost of 20lb of apples is: $7.25
In Franklin’s fruit Orchard,
The cost of 12lb of apples is: $5
We know that,
Where the cost of 1lb of apples is low, Mei Li will go there to buy the apples
Now,
In Annie’s Apple Orchard,
The cost of 1lb of apples = \(\frac{$7.25}{20}\)
= $0.3625
In Franklin’s fruit Orchard,
The cost of 1lb of apples = \(\frac{$5}{12}\)
= $0.4166
So,
The cost of 1lb of apples in Annie’s Orchard < The cost f 1lb of apples in Franklin’s fruit Orchard
Hence, from the above,
We can conclude that Mei Li should go to pick apples from Annie’s Apple Orchard

Construct Arguments
What information provided can be used to support your answer?
Answer:
From the given figure,
The information provided that can be used to support your answer is:
The weight of the apples is inversely proportional to the price of the apples
So,
In Annie’s Apple Orchard, the weight of the apples is high when compared to the weight of the apples in franklin’s fruit Orchard
Hence,
The price of the apples is low in Annie’s Apple Orchard when compared to Franklin’s fruit Orchard

Focus on math practices
Model with Math Which representation did you use to compare prices? Explain why.
Answer:
The relation that is used to compare the prices of apples is:
Weight of the apples ∝ \(\frac{1}{Price of the apples}\)

? Essential Question
How can you compare proportional relationships represented in different ways?
Answer:
To compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.

Try It!
The graph represents the rate at which Marlo makes origami birds for a craft fair. The equation y = 2.5x represents the number of birds, y, Josh makes in x minutes. Who makes birds at a faster rate?

Answer:
It is given that
The graph represents the rate at which Marlo makes origami birds for a craft fair. The equation y = 2.5x represents the number of birds, y, Josh makes in x minutes.
So,
The rate that birds made by Josh = \(\frac{y}{x}\)
= 2.5
Now,
From the graph,
The rate that birds made by Marlo = \(\frac{Time taken to make birds by Marlo}{The number of birds}\)
= \(\frac{40}{8}\) (Here, we can take any value that is present in the graph. For example,\(\frac{20}{4}\), \(\frac{10}{2}\) etc., )
= 5
So,
The rate that birds made by Marlo > The rate that birds made by Josh
Hence, from the above,
We can conclude that Marlo makes birds at a faster rate

Convince Me!
If you were to graph the data for Josh and Marlo on the same coordinate plane, how would the two lines compare?
Answer:
When we graph the data for Josh and Marlo on the same coordinate plane,
We can observe that the two graphs will be the lines that are parallel to each other and the rate of change  of Marlo will be greater than the rate of change of Josh

Try It!
The distance covered by the fastest high-speed train in Japan traveling at maximum speed is represented on the graph. The fastest high-speed train in the United States traveling at maximum speed covers 600 kilometers in 2\(\frac{1}{2}\) hours. Which train has a greater maximum speed? Explain.

Answer:
It is given that
The distance covered by the fastest high-speed train in Japan traveling at maximum speed is represented on the graph. The fastest high-speed train in the United States traveling at maximum speed covers 600 kilometers in 2\(\frac{1}{2}\) hours.
Now,
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
The speed of the fastest high-speed train in the United states = 600 / \(\frac{5}{2}\)
We know that,
2\(\frac{1}{2}\) = \(\frac{5}{2}\)
So,
The speed of the fastest high-speed train in the United states = \(\frac{600 × 2}{5}\)
= 240 kilometers per hour
Now,
From the given graph,
The speed of the fastest high-speed train in Japan = \(\frac{The difference between any two distances from the graph}{The  difference between the values of the time that corresponds to the taken value of distances}\)
= \(\frac{1000 – 650}{3 – 2}\)
= 350 kilometers per hour
So,
The speed of the fastest high-speed train in Japan > The speed of the fastest high-speed train in the United States
Hence, from the above,
We can conclude that the fastest high-speed train in Japan has a maximum speed

KEY CONCEPT
To compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.
The representations below show the rental cost per hour for canoes at three different shops.

Do You Understand?

Question 1.
?Essential Question How can you compare proportional relationships represented in different ways?
Answer:
To compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.

Question 2.
How can you find the unit rate or constant of proportionality for a relationship represented in a graph?
Answer:
In a graph,
The unit rate or constant of proportionality for a relationship is represented by:
\(\frac{The value of y}{The value of x}\) or \(\frac{The difference between any 2 values of y}{The difference between the values of x that is corresponded to the values of x}\)

Question 3.
Generalize Why can you use the constant of proportionality with any representation?
Answer:
We can use the constant of proportionality to find the rate of change between the physical quantities that have a proportional relationship
Ex:
Speed Vs Distance, Speed Vs Time, etc

Do You Know How?

Question 4.
Amanda babysits and Petra does yard work on weekends. The graph relating Amanda’s earnings to the number of hours she babysits passes through the points (0, 0) and (4, 24). The table below relates Petra’s earnings to the number of hours she does yard work.

Who earns more per hour?
Answer:
It is given that
Amanda babysits and Petra does yard work on weekends. The graph relating Amanda’s earnings to the number of hours she babysits passes through the points (0, 0) and (4, 24). The table below relates Petra’s earnings to the number of hours she does yard work.
Now,
The Earnings per hour of Amanda = \(\frac{24 – 0}{4 – 0}\)
= \(\frac{24}{4}\)
= 6
The Earnings per hour of Petra = \(\frac{15}{3}\)
= 5
So,
The Earnings per hour of Amanda > The Earnings per hour of Petra
Hence, from the above,
We can conclude that Amanda earns more

Question 5.
Milo pays $3 per pound for dog food at Pat’s Pet Palace. The graph below represents the cost per pound of food at Mark’s Mutt Market. At which store will Milo pay a lower price per pound for dog food?

Answer:
It is given that
Milo pays $3 per pound for dog food at Pat’s Pet Palace. The graph below represents the cost per pound of food at Mark’s Mutt Market.
So,
Now,
The cost per pound of food at Mark’s Mutt Market = \(\frac{Any value of cost from the given graph}{The value of weight that corresponds to the selected cost}\)
= \(\frac{5}{1}\)
= $5
So,
The cost per pound of food at Pat’s Pet Palace < The cost per pound of food at Mark’s Mutt Market
Hence, from the above,
We can conclude that at Pat’s Pet Palace, Milo will pay a lower price per pound for dog food

Practice & Problem Solving

Leveled Practice For 6 and 7, complete the information to compare the rates.

Question 6.
Sam and Bobby want to know who cycled faster. The table shows the total miles Sam traveled over time. The graph shows the same relationship for Bobby. Who cycled faster.


Find the unit rate (constant of proportionality) for Bobby.
Use () and () to find the constant of proportionality.
The unit rate (constant of proportionality) is
So cycled faster.
Answer:
It is given that
Sam and Bobby want to know who cycled faster. The table shows the total miles Sam traveled over time. The graph shows the same relationship for Bobby.
We know that,
Speed = \(\frac{Distance}{Time}\)
So,
For Sam, from the table,
Speed = \(\frac{20}{2}\) miles per hour
= 10 miles per hour
Now,
For Bobby, from the graph,
Speed = \(\frac{Any value of the distance from the graph}{The value of time that corresponds to the distance that we have taken}\)
= \(\frac{72}{8}\)
= 9 miles per hour
So,
The speed of Sam > The speed of Bobby
Hence, from the above,
We can conclude that Sam cycled faster

Question 7.
Model with Math The equation y = 15x can be used to determine the amount of money, y, Pauli’s Pizzeria makes by selling x pizzas. The graph shows the money Leo’s Pizzeria takes in for different numbers of pizzas sold. Which pizzeria makes more money per pizza?

Pauli’s Pizzeria takes in per pizza.
Leo’s Pizzeria takes in per pizza.
‘s Pizzeria takes in more money per pizza.
Answer:
It is given that
The equation y = 15x can be used to determine the amount of money, y, Pauli’s Pizzeria makes by selling x pizzas. The graph shows the money Leo’s Pizzeria takes in for different numbers of pizzas sold
So,
The money earned by Pauli’s Pizzeria = \(\frac{y}{x}\)
= 15 (From the given equation y = 15x)
Now,
From the given graph,
The money earned by Leo’s Pizzeria = \(\frac{Any value of the amount made from the graph}{The value of pizzas sold that corresponds to the value of the amount that we have considered}\)
= \(\frac{96}{8}\)
= 12
So,
The money earned by Pauli’s Pizzeria > The money earned by Leo’s Pizzeria
Hence, from the above,
We can conclude that Pauli’s Pizzeria takes in more money per pizza

Question 8.
The graph shows the amount of savings over time in Eliana’s account. Lana, meanwhile, puts $50 each week into her savings account. If they both begin with $0, who is saving at the greater rate?

Answer:
It is given that
The graph shows the amount of savings over time in Eliana’s account. Lana, meanwhile, puts $50 each week into her savings account
So,
The amount of savings over time in Lana’s account = \(\frac{Any value of total savings in the graph}{The corresponding value of time to that savings amount}\)
= \(\frac{94}{2}\)
= $47
So,
The amount of savings over time in Elina’s account > The amount of savings over time in Lana’s account
Hence, from the above,
We can conclude that Elina is saving money at a greater rate

Question 9.
Make Sense and Persevere Beth, Manuel, and Petra are collecting sponsors for a walk-a-thon. The equation y = 20x represents the amount of money Beth raises for walking x miles. The table shows the relationship between the number of miles Manuel walks and the amount of money he will raise. Petra will earn $15 for each mile that she walks.
a. In order to compare the proportional relationships, what quantities should you use to find the unit rate?
Answer:
In order to compare the proportional relationships,
The quantities you should use to find the unit rate is:
A) The number of miles walked
B) The amount of money raised for the corresponding number of miles

b. Compare the amount of money raised per mile by the three people.

Answer:
It is given that
Beth, Manuel, and Petra are collecting sponsors for a walk-a-thon. The equation y = 20x represents the amount of money Beth raises for walking x miles. The table shows the relationship between the number of miles Manuel walks and the amount of money he will raise. Petra will earn $15 for each mile that she walks.
So,
The amount of money raised by Beth = \(\frac{y}{x}\)
= $20 (From the equation y = 20x)
Now,
The amount of money raised by Manuel = \(\frac{Any value of the money raised in the table}{The number of miles walked that corresponds to the value of money raised}\)
= \(\frac{$45}{3}\)
= $15
So,
The amount of money raised by Beth > The amount of money raised by Manuel = The amount of money raised by Petra
Hence, from the above,
We can conclude that Beth raised more amount of money when compared to Manuel and Petra

Question 10.
Higher-Order Thinking Winston compares the heights of two plants to see which plant grows more per day. The table shows the height of Plant 1, in centimeters, over 5 days. The graph shows the height of Plant 2, in centimeters, over 10 days. Winston says that since Plant 1 grows 6 cm per day and Plant 2 grows 4 cm per day, Plant 1 grows more per day.

a. Do you agree with Winston? Explain your response.
Answer:
It is given that
Winston compares the heights of two plants to see which plant grows more per day. The table shows the height of Plant 1, in centimeters, over 5 days. The graph shows the height of Plant 2, in centimeters, over 10 days. Winston says that since Plant 1 grows 6 cm per day and Plant 2 grows 4 cm per day, Plant 1 grows more per day.
So,
From the given information,
The height growth of plant 1 > The height growth of plant 2
Hence, from the above,
You can agree with Winston

b. What errors might Winston have made?
Answer:
For plant 1,
The height growth per day = \(\frac{Any value of height}{The value of days correspond to the value of height}\)
= \(\frac{6}{2}\)
= 3 cm
For plant 2,
The height growth per day = \(\frac{Any value of height}{The value of days correspond to the value of height}\)
= \(\frac{4}{2}\)
= 2 cm
But,
It is given that
Winston says that Plant 1 grows 6 cm per day and Plant 2 grows 4 cm per day
But according to the calculation,
Plant 1 grows 3 cm per day and plant 2 grows 2 cm per day
So,
The calculation of the height growth of the plants are the errors made by Winston

Assessment Practice

Question 11.
Ashton, Alexa, and Clara want to know who types the fastest. The equation y = 39x models the rate at which Ashton can type, where y is the number of words typed and x is the time in minutes. The table shows the relationship between words typed and minutes for Alexa. The graph shows the same relationship for Clara. Who types the fastest?

Answer:
It is given that
Ashton, Alexa, and Clara want to know who types the fastest. The equation y = 39x models the rate at which Ashton can type, where y is the number of words typed and x is the time in minutes. The table shows the relationship between words typed and minutes for Alexa. The graph shows the same relationship for Clara.
So,
The rate at which Ashton can type = \(\frac{y}{x}\)
= 39 words per minute (From the equation y = 39x)
The rate at which Alexa can type = \(\frac{Any value of the words typed from the table}{The value of minds corresponds to the words typed}\)
= \(\frac{78}{2}\)
= 39 words per minute
The rate at which Clara can type = \(\frac{Any value of the words typed from the graph}{The value of minds corresponds to the words typed}\)
= \(\frac{78}{2}\)
= 39 words per minute
So,
The rate at which Ashton can type = The rate at which Alexa can type = the rate at which Clara can type
Hence, from the above,
We can conclude that no one is the fastest

Lesson 2.6 Connect Proportional Relationships and Slope

ACTIVITY

Solve & Discuss It!

In the fall, Rashida earns money as a soccer referee for her town’s under-10 soccer league. So far, she has worked 5 games and has been paid $98.50. She will work a total of 14 games this fall. How can Rashida determine how much she will earn refereeing soccer games this fall?

Answer:
It is given that
In the fall, Rashida earns money as a soccer referee for her town’s under-10 soccer league. So far, she has worked 5 games and has been paid $98.50. She will work a total of 14 games this fall.
So,
The amount of money paid for 1 game = \(\frac{The amount of money paid for 5 games}{5}\)
= \(\frac{$98.50}{5}\)
= $19.70
So,
The amount of money paid for 14 games to Rashida = (The total number of games) × (The amount of money paid for 1 game)
= 14 × $19.70
= $275.80
Hence, from the above,
We can conclude that by finding out the money paid to a game for Rashida, Rashida can find total money earned by refereeing soccer games in the fall

Look for Relationships
How is the number of games Rashida works related to her earnings?
Answer:
From the above,
We can observe that Rashida earns more money by refereeing more soccer games
Hence, from the above,
We can conclude that
The number of games Rashida works ∝ The earnings of Rashida

Focus on math practices
Reasoning: How would Rashida’s earnings change if she were paid by the hour instead of by the game?
Answer:
Rashida’s earnings would increase if she were paid by the hour instead of by the game
Example:
From the above,
We can observe that
The money earned by Rashida per game = $17.90
But, if a game will continue for 2 hours and the amount of money that is per game will also be applicable to this situation, then
The amount of money earned by Rashida for this game = $17.90 × 2 = $35.80
Hence, from the above,
We can conclude that Rashida can earn more if she were paid by the hour instead of by the game

? Essential Question
What is the slope?
Answer:
The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as “rise over run” (change in y divided by change in x).
The representation of the slope mathematically is:
Slope = \(\frac{Rise}{Run}\)

Try It!

Jack graphs how far he plans to bike over a 3-day charity ride. Find the slope of the line.


Answer:
It is given that Jack graphs how far he plans to bike over a 3-day charity ride
Now,
From the given graph,
The given points are: (3, 90), and (2, 60)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{60 – 90}{2 – 3}\)
= 30
Hence, from the above,
We can conclude that the slope of the line is: 30

Convince Me!
How do the unit rate and constant of proportionality relate to the slope of a line?
Answer:
The relative steepness of the line is called slope. The slope of a graph is the same as the constant of proportionality of the equation. A line with a steeper slope has a larger value for k.

Try It!
The graph shows the proportions of red and blue food coloring that Taylor mixes to make the purple frosting. What is the slope of the line? Tell what it means in the problem situation.

Answer:
It is given that
The graph shows the proportions of red and blue food coloring that Taylor mixes to make the purple frosting.
Now,
From the given graph,
The given points are: (50, 70), and (25, 35)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{70 – 35}{50 – 25}\)
= \(\frac{35}{25}\)
= \(\frac{7}{5}\)
Hence, from the above,
We can conclude that
For every 7 parts of red food coloring, we have to mix 5 parts of blue food coloring to make the purple frosting

KEY CONCEPT

Slope is the measure of the steepness of a line. It represents the ratio of the rise (that is, the vertical distance) to the run (the horizontal distance) between two points on the line. In proportional relationships, slope is the same as the unit rate and constant of proportionality.

Do You Understand?

Question 1.
? Essential Question
What is the slope?
Answer:
The slope of a line is a measure of its steepness. Mathematically, the slope is calculated as “rise over run” (change in y divided by change in x).
The representation of the slope mathematically is:
Slope = \(\frac{y2  – y1}{x2 – x1}\) (or) Sloe = \(\frac{Rise}{Run}\)

Question 2.
Reasoning How is the slope related to a unit rate?
Answer:
The slope is the unit rate, which is the coefficient of x. For a table, the change in y divided by the change in x is the unit rate or slope.

Question 3.
Look for Relationships Why is the slope between any two points on a straight line always the same?
Answer:
The ratio of the rise over run describes the slope of all straight lines. This ratio is constant between any two points along a straight line, which means that the slope of a straight line is constant, too, no matter where it is measured along the line.

Do You Know How?

Question 4.
What is the slope of the line?

Answer:
The given graph is:

We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Price ($)}{Grapes (lb)}\)
So,
The slope of the line = \(\frac{6}{2}\)
= 3
Hence, from the above,
We can conclude that the slope of the line is: 3

Question 5.
The scale of a model airplane is shown in the graph.
a. Find the slope of the line using \(\frac{y2  – y1}{x2 – x1}\)

Answer:
The given graph is:

Now,
From the given graph,
The given points are: (6, 10), and (3, 5)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{5 – 10}{3 – 6}\)
= \(\frac{5}{3}\)
Hence, from the above,
We can conclude that the slope of the line is: \(\frac{5}{3}\)

b. What does the slope mean in the problem situation?
Answer:
From part (a),
The slope is: \(\frac{5}{3}\)
So,
From the above slope,
We can conclude that for every 3 cm, the model airplane can fly 5 feet

Practice & Problem Solving

Leveled Practice in 6 and 7, find the slope of each line.

Question 6.
The graph shows the number of soda bottles a machine can make over time. Use the two points shown to find the number of soda bottles the machine can make per minute.


The machine can make soda bottles each minute.
Answer:
It is given that
The graph shows the number of soda bottles a machine can make over time
Now,
The given graph is:

Now,
From the given graph,
The given points are: (6, 150), and (2, 50)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y2  – y1}{x2 – x1}\)
So,
The slope of the line = \(\frac{50 – 150}{2 – 6}\)
= \(\frac{100}{4}\)
= 25
Hence, from the above,
We can conclude that the machine can make 25 soda bottles each minute

Question 7.
Find the slope of the line.

Answer:
The given graph is:

We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Items}{Time in min}\)
So,
The slope of the line = \(\frac{50}{10}\)
= 5
Hence, from the above,
We can conclude that the slope of the line is: 5

Question 8.
Reasoning How can you find the slope of the line that passes through the points (0,0) and (2, 4)? Explain.
Answer:
The given points are: (0, 0), and (2, 4)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y₂  – y₁}{x₂ – x₁}\)
So,
The slope of the line = \(\frac{4 – 0}{2 – 0}\)
= \(\frac{4}{2}\)
= 2
Hence, from the above,
We can conclude that the slope of the line is: 2

Question 9.
The points (2.1, -4.2) and (2.5, -5) form a proportional relationship. What is the slope of the line that passes through these two points?
Answer:
It is given that the points (2.1, -4.2) and (2.5, -5) form a proportional relationship
Now,
The given points are: (2.1, -4.2), and (2.5, -5)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y₂  – y₁}{x₂ – x₁}\)
So,
The slope of the line = \(\frac{-5 + 4.2}{2.5 – 2.1}\)
= \(\frac{-0.8}{0.4}\)
= -2
Hence, from the above,
We can conclude that the slope of the line that passes through the given points is: -2

Question 10.
Find the slope of the line.

Answer:
The given graph is:

Now,
From the graph,
We can observe that
The given points are: (-3, 7), and (-1, 2)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y₂  – y₁}{x₂ – x₁}\)
So,
The slope of the line = \(\frac{2 – 7}{-1 + 3}\)
= \(\frac{-5}{2}\)
= –\(\frac{5}{2}\)
Hence, from the above,
We can conclude that the slope of the line that passes through the given points is: –\(\frac{5}{2}\)

Question 11.
The graph shows the number of Calories Natalia burned while running.
a. What is the slope of the line?

Answer:
The given graph is:

We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Calories}{Time in min}\)
So,
The slope of the line = \(\frac{70}{7}\)
= 10
Hence, from the above,
We can conclude that the slope of the line is: 10

b. What does the slope tell you?
Answer:
From part (a),
We can observe that
The slope of the line is: 10
So,
From the given slope,
We can conclude that Natalia burns 10 calories per minute while running

Question 12.
Critique Reasoning A question on a test provides this graph and asks students to find the speed at which the car travels. Anna incorrectly says that the speed of the car is \(\frac{1}{64}\) mile per hour.
a. What is the speed of the car?

Answer:
The given graph is:

We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\)
= \(\frac{Distance in miles}{Time in hours}\)
So,
The slope of the line = \(\frac{256}{4}\)
= 64
Hence, from the above,
We can conclude that the speed of the car is: 64 miles per hour

b. What error might Anna have made?
Answer:
From part (a),
We can observe that the speed of the car is: 64 miles per hour
Bt,
According to Anna,
The speed of the car is: \(\frac{1}{64}\) miles per hour
So,
The error made by Anna is that she takes the slope in the form of \(\frac{x}{y}\) but the actual form of the slope is \(\frac{y}{x}\)

Question 13.
Higher-Order Thinking You use a garden hose to fill a wading pool. If the water level rises 11 centimeters every 5 minutes and you record the data point of (10, y), what is the value of y? Use slope to justify your answer.

Answer:
It is given that
You use a garden hose to fill a wading pool. If the water level rises 11 centimeters every 5 minutes and you record the data point of (10, y)
We know that,
Slope = \(\frac{Rise}{Run}\)
So,
From the given information,
We can write the slope as:
Slope = \(\frac{11}{5}\)
Now,
Compare the given point with (x, y)
So,
The slope of the line = \(\frac{y}{x}\)
= \(\frac{y}{10}\)
So,
\(\frac{y}{10}\) = \(\frac{11}{5}\)
Multiply with 10 on both sides
So,
y = \(\frac{11 × 10}{5}\)
y = 22
Hence, from the above,
We can conclude that the value of y is: 22

Assessment Practice

Question 14.
The points (15, 21) and (25, 35) form a proportional relationship.
a. Find the slope of the line that passes through these points.
Answer:
It is given that the points (15, 21) and (25, 35) form a proportional relationship.
Now,
The given points are: (15, 21), and (25, 35)
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope = \(\frac{Rise}{Run}\) = \(\frac{y₂  – y₁}{x₂ – x₁}\)
So,
The slope of the line = \(\frac{35 – 21}{25 – 15}\)
= \(\frac{14}{10}\)
= \(\frac{7}{5}\)
Hence, from the above,
We can conclude that the slope of the line that passes through the given points is: \(\frac{7}{5}\)

b. Which graph represents this relationship?

Answer:
We know that,
The representation of the equation when two points form  a proportionate relationship is:
y = kx
and the line have to pass through the origin i.e., (0, 0)
So,
From the given graphs,
The graphs B and C have the possibility to become the graph of the given points
Now,
We know that,
Slope = \(\frac{Rise}{Run}\)
From graph B,
Slope = \(\frac{42}{30}\)
= \(\frac{7}{5}\)
From graph C,
Slope = \(\frac{30}{42}\)
= \(\frac{5}{7}\)
Hence, from the above,
We can conclude that the graph B represents the given relationship

Lesson 2.7 Analyze Linear Equations: y = mx

ACTIVITY

Explore It!

A group of college students developed a solar-powered car and entered it in a race. The car travels at a constant speed of 100 meters per 4 seconds.

A. What representation can show the distance the car will travel over time?
Answer:
It is given that
A group of college students developed a solar-powered car and entered it in a race. The car travels at a constant speed of 100 meters per 4 seconds.
We know that,
Speed = \(\frac{Distance}{Time}\)
It is given that speed is constant
So,
Distance ∝ Time
So,
The greater the distance, the greater the time
Hence, from the above,
We can conclude that
The representation that can show the distance the car will travel over time is:
Distance ∝ Time

B. What expression can show the distance the car will travel over time?
Answer:
From part (a),
We can observe that
Distance ∝ Time (Since the speed is constant)
Hence,
The expression that can show the distance the car will travel over time is:
Distance = k (Time)
Where,
k is a constant

C. Compare the representation and the expression. Which shows the distance traveled over time more clearly? Explain.
Answer:
From part (a),
The representation that can show the distance traveled over time is:
Distance ∝ Time
The expression that can show the distance traveled over time is:
Distance = k (Time)
Now,
From the representation and the expression,
We can observe that the expression shows the distance traveled over time more clearly because for any value of distance and time, the value of the expression is constant
Hence, from the above,
We can conclude that the expression shows the distance traveled over time more clearly

Focus on math practices
Be Precise How would the representation or expression change if the speed was converted to miles per minute?
Answer:
From part (a),
The representation is:
Distance ∝ Time
The expression is:
Distance = k (Time)
Now,
Even if the speed was converted to miles per minute, there will be no change in the representation and the expression because miles per minute is a unit of speed and it won’t affect the overall situation of the representation and the expression

? Essential Question
How does slope relate to the equation for a proportional relationship?
Answer:
The steepness of the slope for directly proportional relationships increases as the value of the constant m (y = mx) increases.

Try It!
Write an equation to describe the relationship shown in the graph.
. The equation of the line is y = x.

Answer:
The given graph is:

Now,
From the given graph,
The points are: (3, 60), and (4, 80) [We can take any 2 ordered pairs from the graph like (0, 0), and (1, 20); (2, 40), and (3, 60), etc]
Now,
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope can be represented as “m”
So,
m = \(\frac{y₂  – y₁}{x₂ – x₁}\)
So,
m = \(\frac{80 – 60}{4 – 3}\)
= \(\frac{20}{1}\)
= 20
We know that,
The equation of the line is:
y = mx
Hence, from the above,
We can conclude that
The equation of the line is: y = 20x

Convince Me!
How do the equations y = mx and y = kx compare?
Answer:
We can compare y = kx to the slope-intercept form of a line, y = mx + b. We can see that y = kx is a linear equation with slope k and y-intercept 0. This tells us that the graph of a direct variation is a line that passes through the origin, point (0,0).

Try It!
a. Write the equation of the line.

Answer:
The given graph is:

Now,
From the given graph,
The points are: (10, 4), and (-10, -4)
Now,
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope can be represented as “m”
So,
m = \(\frac{y₂  – y₁}{x₂ – x₁}\)
So,
m = \(\frac{-4 – 4}{-10 – 10}\)
= \(\frac{-8}{-20}\)
= \(\frac{2}{5}\)
We know that,
The equation of the line is:
y = mx
So,
y = \(\frac{2}{5}\)x
Multiply with 5 on both sides
So,
5y = 2x
Hence, from the above,
We can conclude that
The equation of the line is: 5y = 2x

b. Graph the line y = -3x.

Answer:
The given equation is:
y = -3x
Hence,
The representation of the given equation in the coordinate plane is:

KEY CONCEPT

The equation for a proportional relationship is y = mx where m represents the slope of the line.

Do You Understand?

Question 1.
Essential Question How does slope relate to the equation for a proportional relationship?
Answer:
The steepness of the slope for directly proportional relationships increases as the value of the constant m (y = mx) increases.

Question 2.
Look for Relationships What do the graphs of lines in the form y = mx have in common? How might they differ?
Answer:
The graphs of lines in the form y = mx are all straight lines that pass through the origin

Question 3.
Use Structure The table below shows the distance a train traveled over time. How can you determine the equation that represents this relationship?

Answer:
It is given that
The table below shows the distance a train traveled over time.
Now,
Verify whether \(\frac{Distance}{Time}\) is constant or not
Now,
From the given table,
For 25 m and 2s,
\(\frac{Distance}{Time}\) = \(\frac{25}{2}\)
For 50m and 4s,
\(\frac{Distance}{Time}\) = \(\frac{50}{4}\) = \(\frac{25}{2}\)
Since,
\(\frac{Distance}{Time}\) is constant
Speed is also constant
So,
The representation of the equation that describes the given relationship is:
Distance = k (Time)
So,
y = mx [ Compare the above equation with y = mx ]
Where
m is a constant slope
So,
y = \(\frac{25}{2}\)x
2y = 25x
Hence, from the above,
We can conclude that the representation of the equation that represents the given situation is: 2y = 25x

Do You Know How?

Question 4.
The relationship between a hiker’s elevation and time is shown in the graph.

a. Find the constant of proportionality of the line. Then find the slope of the line.
Answer:
It is given that
The relationship between a hiker’s elevation and time is shown in the graph.
Now,
We know that,
The constant of proportionality and the slope are the same
So,
Slope of the line (m) = \(\frac{y}{x}\)
So,
From the given graph,
\(\frac{y}{x}\) = \(\frac{120}{4}\)
= 30
So,
m = 30
Hence, from the above,
We can conclude that the slope of the line is: 30

b. Write the equation of the line.
Answer:
We know that,
The equation of the line is:
y = mx
From part (a),
m = 30
Hence, from the above,
We can conclude that the equation of the line is: y = 30x

Question 5.
Graph the equation y = –\(\frac{1}{2}\)x.

Answer:
The given equation is:
y = –\(\frac{1}{2}\)x
Hence,
The representation of the given equation in the coordinate plane is:

Practice & Problem Solving

Question 6.
Leveled Practice Resting heart rate is a measure of how fast the heart beats when a person is not performing physical activity. The graph shows the number of heartbeats over time for a given person.
a. Use two sets of coordinates to write an equation to describe the relationship.


Answer:
It is given that
Resting heart rate is a measure of how fast the heart beats when a person is not performing physical activity. The graph shows the number of heartbeats over time for a given person.
Now,
From the given graph,
The points are: (3, 210), and (4, 280)
Now,
Compare the given points with (x₁, y₁), and (x₂, y₂)
We know that,
Slope can be represented as “m”
So,
m = \(\frac{y₂  – y₁}{x₂ – x₁}\)
So,
m = \(\frac{280 – 210}{4 – 3}\)
= \(\frac{70}{1}\)
= 70
We know that,
The equation of the line is:
y = mx
Hence, from the above,
We can conclude that
The equation of the line that describes the given situation is: y = 70x

b. Interpret the equation in words.
The heart’s resting heart rate is beats each minute.
Answer:
From part (a),
The equation of the line that describes the given situation is: y = 70x
Hence, from the above,
We can conclude that the heart’s resting heart rate is 70 beats each minute

Question 7.
Model with Math The graph relates the number of gallons of white paint to the number of gallons of red paint Jess used to make the perfect pink. Write an equation that describes the relationship.

Answer:
It is given that
The graph relates the number of gallons of white paint to the number of gallons of red paint Jess used to make the perfect pink.
Now,
The given graph is:

Now,
Slope of the given line (m) = \(\frac{y}{x}\)
m = \(\frac{4}{1}\)
m = 4
We know that,
The equation of the line is:
y = mx
Hence, from the above,
We can conclude that the equation of the line that represents the given situation is: y = 4x

Question 8.
Critique Reasoning Franco made this graph to show the equation y = -x. Is the graph correct? Explain.

Answer:
The given graph is:

Now,
We know that,
Slope of the line (m) = \(\frac{y}{x}\)
m = \(\frac{4}{4}\)
m = 1
We know that,
The equation of the line is:
y = mx
So,
The equation of the line is:
y = x
But,
Franco made this graph to show the equation y = -x
Hence, from the above,
We can conclude that the graph of Franco is not correct

Question 9.
The graph shows a proportional relationship between the variables x and y.
a. Write an equation to model the relationship.
b. Reasoning Explain how you know if an equation or a graph represents a proportional relationship.

Answer:
a.
The given graph is:

We know that,
The slope of the line (m) = \(\frac{y}{x}\)
= \(\frac{96}{8}\)
= 12
We know that,
The equation of the line is:
y = mx
So,
The equation of the line to the given relationship is:
y = 12x
Hence, from the above,
We can conclude that the equation of the line that represents the given situation is: y = 12x

b.
If the relationship between two quantities is a proportional relationship, this relationship can be represented by the graph of a straight line through the origin with a slope equal to the unit rate. For each point (x, y) on the graph, ž is equal to k, where k is the unit rate.

Question 10.
Model with Math Graph the equation y = -5x on the coordinate plane.

Answer:
The given equation is:
y = -5x
Hence,
The representation of the given equation in the coordinate plane is:

Question 11.
Graph the equation y = \(\frac{3}{5}\)x on the coordinate plane.

Answer:
The given equation is:
y = \(\frac{3}{5}\)x
Hence,
The representation of the given equation in the coordinate plane is:

Question 12.
Higher-Order Thinking A movie theater sends out a coupon for 70% off the price of a ticket.
a. Write an equation for the situation, where y is the price of the ticket with the coupon and x is the original price.

Answer:
It is given that
A movie theater sends out a coupon for 70% off the price of a ticket.
So,
The proportionality constant of the given situation = \(\frac{70}{100}\)
= \(\frac{7}{10}\)
We know that,
Proportionality constant = Slope
So,
Slope (m) = \(\frac{7}{10}\)
We know that,
The equation of the line is:
y = mx
So,
y = \(\frac{7}{10}\)
10y = 7x
Hence, from the above,
We can conclude that the equation of the line for the given situation is: 10y = 7x

b. Graph the equation and explain why the line should only be in the first quadrant.
Answer:
From part (a),
The equation of the line is:
10y = 7x
So,
The representation of the given equation in the coordinate plane is:

From the graph,
We can observe that
The graph should only be in 1st quadrant because the values of x and y are both positive

Assessment Practice

Question 13.
An equation and a graph of proportional relationships are shown. Which has the greater unit rate? y = \(\frac{47}{2}\)x

Answer:
The given graph is:

From the given graph,
Slope (m) = \(\frac{y}{x}\)
= \(\frac{282}{6}\)
= 47
Now,
The given equation is:
y = \(\frac{47}{2}\)x
So,
Slope (m) = \(\frac{y}{x}\)
= \(\frac{47}{2}\)
Now,
When we compare the rates or slopes,
47 > \(\frac{47}{2}\)
Hence, from the above,
We ca conclude that the unit rate of the graph is greater than the unit rate of the equation

Question 14.
Car X travels 186 miles in 3 hours.
PART A Write the equation of the line that describes the relationship between distance and time.
Answer:
It is given that car X travels 186 miles in 3 hours.
Now,
We know that,
Speed = \(\frac{Distance}{Time}\)
We know that,
The equation of the line is:
y = mx
Where,
m = \(\frac{Distance}{Time}\)
So,
The equation of the line is:
y = \(\frac{186}{3}\)x
y = 62x
Hence, from the above,
We can conclude that the equation of the line that descries the relationship between distance and time is:
y = 62x

PART B Which graph represents the relationship between distance and time for Car X?

Answer:
From part (a),
The equation of the line that describes the relationship between distance and time is:
y = 62x
Where,
62 —-> The value of \(\frac{y}{x}\) (or) m
So,
From the above graphs,
We can observe that,
m = 62 is possible from graphs C and D
But,
We know that,
The equation y = mx passes through the origin
Hence, from the above,
We can conclude that the graph C represents the relationship between distance and time for car X

Lesson 2.8 Understand the y-Intercept of a Line

Solve and Discuss It!

Eight-year-old Alex is learning to ride a horse. The trainer says that a horse ages 5 years for every 2 human years. The horse is now 50 years old in human years. How can you determine the age of the horse, in human years, when Alex was born?

Answer:
It is given that
Eight-year-old Alex is learning to ride a horse. The trainer says that a horse ages 5 years for every 2 human years. The horse is now 50 years old in human years.
So,
When Alex is 8 years old,
The age of the horse in human years is: 50 years
Now,
For every 2 human years, the horse ages 5 years
So,
So,
For Alex,
The number of times his age increases = \(\frac{8}{2}\)
= 4 times
So,
The increase in the age of the horse when Alex is 8 years old = 5 × 4 = 20 years
So,
The age of the horse when Alex born = The present age of the horse – The increased age of the horse
= 50 – 20
= 30 years
Hence, from the above,
We can conclude that the age of the horse when Alex is born is: 30 years

Focus on math practices
Use Structure A veterinarian says that cat ages 8 years for every 2 human years. If a cat is now 64 years old in cat years, how old is the cat in human years?
Answer:
It is given that
A veterinarian says that a cat ages 8 years for every 2 human years.
Now,
Let the age of the cat in human years be x
So,
\(\frac{The age of the cat in cat years}{The age of the cat in human years}\) = \(\frac{The increase of the age of the cat for the increase of human years}{The increase of the age of human for the increase of human years}\)
\(\frac{64}{x}\) = \(\frac{8}{2}\)
Divide by 64 into both sides
So,
\(\frac{64}{x × 64}\) = \(\frac{8}{2 × 64}\)
\(\frac{1}{x}\) = \(\frac{1}{16}\)
x = 16 years
Hence, from the above,
We can conclude that the age of cat in human years is: 16 years

? Essential Question
What is the y-intercept and what does it indicate?
Answer:
The slope and y-intercept values indicate characteristics of the relationship between the two variables x and y. The slope indicates the rate of change in y per unit change in x. The y-intercept indicates the y-value when the x-value is 0.

Try It!
Prices for a different bowling alley are shown in the graph. How much does this bowling alley charge for shoe rental? The line crosses the y-axis as
The y-intercept is

Answer:
It is given that
The prices for a different bowling alley are shown in the graph
So,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the given graph,
The given line crosses the y-axis at (0, 3)
We know that,
The y-intercept is the value of y when the value of x is 0
Hence, from the above,
We can conclude that
The given passes through (0, 3)
The y-intercept is: 3

Convince Me!
In these examples, why does the y-intercept represent the cost to rent bowling shoes?
Answer:
In this example,
From the slope,
We can determine the cost of each game
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept and that y-intercept is the cost to rent bowling shoes because the cost won’t ever be zero

Try It!
What is the y-intercept of each graph? Explain.

Answer:
Let the given graphs be named as graph A and graph B respectively
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From graph A,
The y-intercept is: (0, 2)
From graph B,
The y-intercept is: (0, -0.5)

KEY CONCEPT
The y-intercept is the y-coordinate of the point on a graph where the line crosses the y-axis.
When the line crosses through the origin, the y-intercept is 0.
When the line crosses above the origin, the y-intercept is positive.
When the line crosses below the origin, the y-intercept is negative.

Do You Understand?

Question 1.
? Essential Question What is the y-intercept and what does it indicate?
Answer:
The slope and y-intercept values indicate characteristics of the relationship between the two variables x and y. The slope indicates the rate of change in y per unit change in x. The y-intercept indicates the y-value when the x-value is 0.

Question 2.
Look for Relationships Chelsea graphs a proportional relationship. Bradyn graphs a line that passes through the origin. What do you know about the y-intercept of each student’s graph? Explain your answer.
Answer:
It is given that
Chelsea graphs a proportional relationship. Bradyn graphs a line that passes through the origin
So,
From the given situation,
We can observe that
The graph of Chelsea may pass through the origin or may not pass through the origin i.e., the y-intercept may be zero, positive, or negative
The graph of Braydon passes through the origin i.e., the y-intercept is zero

Question 3.
Generalize When the y-intercept is positive, where does the line cross the y-axis on the graph? When it is negative?
Answer:
When the y-intercept is positive, the line crosses above the origin,
When the y-intercept is negative, the line crosses below the origin

Do You Know How?

Question 4.
What is the y-intercept shown in the graph?

Answer:
The given graph is:

From the given graph,
We can observe that the line passes through the origin
Hence, from the above,
We can conclude that the value of the y-intercept is: 0

Question 5.
The graph shows the relationship between the remaining time of a movie and the amount of time since Kelly hit “play.” What is the y-intercept of the graph and what does it represent?

Answer:
It is given that
The graph shows the relationship between the remaining time of a movie and the amount of time since Kelly hit “play.”
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that the line crosses the y-axis at (0, 1.8)
Hence, from the above,
We can conclude that the y-intercept of the graph is: 1.8
The y-intercept represents the remaining time of a movie in the given situation

Practice & Problem Solving

Question 6.
Leveled Practice Find the y-intercept of the line. The y-intercept is the point where the graph crosses the -axis.
The line crosses the y-axis at the point
The y-intercept is

Answer:
The given graph is:

Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that
The line crosses the y-axis at the point (0, 7)
Hence, from the above,
We can conclude that
The y-intercept is the point where the graph crosses the y-axis
The y-intercept for the given graph is: 8

Question 7.
Find the y-intercept of the graph.

Answer:
The given graph is:

Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that
The line crosses the y-axis at the point (0, -4)
Hence, from the above,
We can conclude that
The y-intercept for the given graph is: -4

Question 8.
Find the y-intercept of the graph.

Answer:
The given graph is:

Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
Now,
From the graph,
We can observe that
The equation of the line is:
y = kx
From the above equation,
We can say that the line passes through the origin
So,
The line crosses the y-axis at the point (0, 0)
Hence, from the above,
We can conclude that
The y-intercept of the given graph is: 0

Question 9.
The graph represents the height y, in meters, of a hot air balloon x minutes after beginning to descend. How high was the balloon when it began its descent?

Answer:
It is given that
The graph represents the height y, in meters, of a hot air balloon x minutes after beginning to descend
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that the line crosses the y-axis at (0, 80)
The y-intercept of the graph gives us information about the height of the balloon when it began its descent
Hence, from the above,
We can conclude that the height of the balloon when it began its descent is: 80 m

Question 10.
Model with Math The graph represents the amount of gasoline in a canister after Joshua begins to fill it at a gas station pump. What is the y-intercept of the graph and what does it represent?

Answer:
It is given that
The graph represents the amount of gasoline in a canister after Joshua begins to fill it at a gas station pump.
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
Now,
From the graph,
We can observe that the line passes through the origin
So,
The line crosses the y-axis at the point (0, 0)
Hence, from the above,
We can conclude that
The y-intercept of the given graph is: 0
The y-intercept of the given graph represents the amount of gas in gallons at the starting time

Question 11.
The line models the temperature on a certain winter day since sunrise.
a. What is the y-intercept of the line?

Answer:
It is given that
The line models the temperature on a certain winter day since sunrise.
Now,
We know that,
y-coordinate of the point where the line crosses the y-axis is the y-intercept
So,
From the graph,
We can observe that the line crosses the y-axis at (0, 4)
Hence, from the above,
We can conclude that the y-intercept of the given line is: 4

b. What does the y-intercept represent?
Answer:
The y-intercept of the graph gives us information about the starting temperature on a certain winter day at sunrise

Question 12.
Higher-Order Thinking Your friend incorrectly makes this graph as an example of a line with a y-intercept of 3.
a. Explain your friend’s possible error.

The given graph is:

Now,
From the given graph,
We can observe that the line crosses the y-axis at: (0, 4)
So,
The y-intercept of the graph is: 4
But,
Your friend incorrectly makes this graph as an example of a line with a y-intercept of 3.
Hence, from the above,
We can conclude that the y-intercept of the given graph is 4 but not 3

b. Draw a line on the graph that does represent a y-intercept of 3.
Answer:
Let the equation with the y-intercept of 3 is:
y = x + 3
Hence,
The representation of the graph that does represent a y-intercept of 3 in the coordinate plane is:

Assessment Practice

Question 13.
For each graph, draw a line through the point such that the values of the x-intercept and y-intercept are additive inverses.

Answer:
Let the graphs be named as graph A and graph B respectively
Now,
The given graphs are:

So,
From graph A,
We can observe that the x-intercept is 3 and the y-intercept is 3
We know that,
The “Additive inverse” of a number ‘a’ is the number that, when added to ‘a’, yields zero. This number is also known as the opposite (number), sign change, and negation.
So,
The additive inverses of the x-intercept and y-intercept are: (-3, -3)
From graph B,
We can observe that the x-intercept is -3 and the y-intercept is -3
We know that,
The “Additive inverse” of a number ‘a’ is the number that, when added to ‘a’, yields zero. This number is also known as the opposite (number), sign change, and negation.
So,
The additive inverses of the x-intercept and y-intercept are: (3, 3)
Hence,
The representation of the additive inverses of the x and y-intercepts in the coordinate plane is:

Question 14.
Which statements describe the graph of a proportional relationship? Select all that apply.
The y-intercept is always at the point (0, 1).
The line always crosses the y-axis at (0, 0).
The y-intercept is 0.
The y-intercept is 1.
The line does NOT cross the y-axis.
Answer:
Let the options be named as A, B, C, D, and E respectively
Now,
We know that,
The representation of the proportional relationship is:
y = kx
So,
From the equation,
We can say that the equation passes through the origin and the y-intercept is 0
Hence, from the above,
We can conclude that options B and C describes the proportional relationship

Lesson 2.9 Analyze Linear Equations: y = mx + b

ACTIVITY

Explain It!

Xiu and Jon take the tram from the base camp to the mountain summit. After about six and a half minutes in the tram, Jon says, “Cool! We are a mile above sea level.” Xiu says, “We passed the one-mile mark a couple of minutes ago.”

A. Construct an argument to defend Xiu’s statement.

B. What mistake could Jon have made? Explain.
Answer:

Focus on math practices
Reasoning Can you use the equation y = mx to represent the path of the tram? Is there a proportional relationship between x and y? Explain.

? Essential Question
What is the equation of a line for a nonproportional relationship?
Answer:
Linear equations can be written in the form y = mx + b. When b ≠ 0, the relationship between x and y is nonproportional.

Try It!
Write a linear equation in slope-intercept form for the graph shown.
The y-intercept of the line is


Answer:
The given graph is:

From the given graph,
We can observe that,
The points are: (8, 8), and (4, 5)
Compare the given points with (x₁, y₁), (x₂, y₂)
We know that,
The y-intercept of the line is the point that crosses the y-axis
So,
From the given graph,
The y-intercept is: 2
We know that,
The linear equation in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
m = \(\frac{y₂ – y₁}{x₂ – x₁}\)
= \(\frac{5 – 8}{4 – 8}\)
= \(\frac{3}{4}\)
So,
The linear equation in the slope-intercept form is:
y = \(\frac{3}{4}\)x + 2
y = \(\frac{3x + 8}{4}\)
4y = 3x + 8
Hence, from the above,
We can conclude that the linear equation in the slope-intercept form is:
4y = 3x + 8

Convince Me!

What two values do you need to know to write an equation of a line, and how are they used to represent a line?
Answer:
To write an equation of a line in the slope-intercept form,
The two values you need to know are:
A) Slope of a line and it is represented as “m”
B) The y-intercept of a line and is represented as “c”

KEY CONCEPT

The equation of a line that represents a nonproportional relationship can be written in slope-intercept form, y = mx + b, where m is the slope of the line and b is the y-intercept.

Do You Understand?

Question 1.
? Essential Question What is the equation of a line for a nonproportional relationship?
Answer:
Linear equations can be written in the form y = mx + b. When b ≠ 0, the relationship between x and y is nonproportional.

Question 2.
Use Structure The donations by a restaurant to a certain charity, y, will be two-fifths of its profits, x, plus $50. How can you determine the equation in slope-intercept form that shows the relationship between x and y without graphing the line?
Answer:
It is given that
The donations by a restaurant to a certain charity, y, will be two-fifths of its profits, x, plus $50.
So,
Donations to a certain charity by a restaurant = The part of the profits of a restaurant + $50
y = \(\frac{2}{5}\)x + $50
Compare the above equation with
y = mx + c
Where,
m is the slope of a line
c is the y-intercept of a line
So,
When we compare the equation,
The slope of a line is (m): \(\frac{2}{5}\)
The y-intercept of a line is (c) : $50

Question 3.
Be Precise Priya will graph a line with the equation y = \(\frac{3}{4}\)x – 4. She wants to know what the line will look like before she graphs the line. Describe the line Priya will draw, including the quadrants the line will pass through.
Answer:
It is given that
Priya will graph a line with the equation y = \(\frac{3}{4}\)x – 4. She wants to know what the line will look like before she graphs the line.
Now,
Compare the given equation with
y = mx + c
Where,
m is the slope of the line
c is the y-intercept
So,
By comparing,
We get,
m = \(\frac{3}{4}\)
c = -4
Now,
From the y-intercept,
We can say that the y-intercept lies below the origin i.e., in the 3rd quadrant
From the slope of the line,
We can say that the value of m lies in the 1st quadrant
Hence, from the above,
We can conclude that the line drawn by Priya will be in the 4th quadrant for the above values of c and m

Do You Know How?

Question 4.
Chrissie says the equation of the line shown on the graph is y = \(\frac{1}{2}\)x – 5. George says that the equation of the line is y = \(\frac{1}{2}\)x + 5. Which student is correct? Explain.

Answer:
It is given that
Chrissie says the equation of the line shown on the graph is y = \(\frac{1}{2}\)x – 5. George says that the equation of the line is y = \(\frac{1}{2}\)x + 5.
Now,
The given graph is

From the given graph,
The y-intercept is: 5
Now,
When we observe the given two equations,
The slope is the same and the y-intercepts are different and the correct y-intercept must be 5
Hence, from the above,
We can conclude that George is correct

Question 5.
Fara wants to rent a tent for an outdoor celebration. The cost of the tent is $500 per hour, plus an additional $100 set-up fee.
a. Draw a line to show the relationship between the number of hours the tent is rented, x, and the total cost of the tent, y.

Answer:
It is given that
Fara wants to rent a tent for an outdoor celebration. The cost of the tent is $500 per hour, plus an additional $100 set-up fee.
Now,
The total cost of the rent = The cost of the rent per hour + Additional set-up fee
So,
y = 500x + 100
Hence,
The representation of the above equation in the coordinate plane is:

b. What is the equation of the line in slope-intercept form?
Answer:
We know that,
The total cost of the rent = The cost of the rent per hour + Additional set-up fee
So,
y = 500x + 100
Where,
x is the number of hours
The above equation is in the form of
y = mx + c
Which is the slope-intercept form of the equation
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = 500x + 100

Practice & Problem Solving

Question 6.
Leveled Practice What is the graph of the equation y = 2x + 4?
The y-intercept is , which means the line crosses the y-axis at the point (). Plot this point.
The slope of the line is positive, so it goes from left to right.
Start at the y-intercept. Move up , and then move right
You are now at the point (). Plot this point. Draw a line to connect the two points.

Answer:
The given equation is:
y = 2x + 4
So,
The representation of the given equation in the coordinate plane is:

Compare the given equation with
y = mx + c
Wher,
m is the slope of a line
c is the y-intercept of a line
So,
The y-intercept of the given graph is 4 which means the line crosses the y-axis at the point (0, 4)
The slope of the line is positive, so it goes up from left to right.
Start at the y-intercept. Move up 2 units, and then move right 2 units
So,
You are now at the point (3, 10).

Question 7.
Write an equation for the line in slope-intercept form.

Answer:
The given graph is:

We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
From the given graph,
We can observe that the y-intercept is: -3
Now,
The given points from the graph to find the slope are: (-2, -2), and (4, -5)
Now,
SLope (m) = \(\frac{-5  – (-2)}{4 – (-2)}\)
m = \(\frac{-3}{6}\)
m = –\(\frac{1}{2}\)
So,
The equation of the line in the slope-intercept form is:
y = –\(\frac{1}{2}\)x – 3
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = –\(\frac{1}{2}\)x – 3

Question 8.
Write an equation for the line in slope-intercept form.

Answer:
The given graph is:

We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
From the given graph,
We can observe that the y-intercept is: 4
Now,
The given points from the graph to find the slope are: (1, 1), and (0, 4)
Now,
SLope (m) = \(\frac{4 – 1}{0 – 1}\)
m = \(\frac{3}{-1}\)
m = -3
So,
The equation of the line in the slope-intercept form is:
y = -3x + 4
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = -3x + 4

Question 9.
The line models the cost of renting a kayak. Write an equation in slope-intercept form for the line, where x is the number of hours the kayak is rented and y is the total cost of renting the kayak.

Answer:
It is given that
The line models the cost of renting a kayak
where,
x is the number of hours the kayak is rented and y is the total cost of renting the kayak.
Nw,
The given graph is:

From the given graph,
We can observe that
The y-intercept of the graph is: 5
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
From the given graph,
The points to find the graph are: (3, 40), and (2, 30)
So,
Slope (m) = \(\frac{30 – 40}{2 – 3}\)
= 10
So,
The equation of the line in the slope-intercept form is:
y = mx + c
So,
y = 10x + 5
Hence, from the above,
We can conclude that
The equation of the line in the slope-intercept form is:
y = 10x + 5

Question 10.
Graph the equation y = 3x – 5.

Answer:
The given equation is:
y = 3x – 5
Hence,
The representation of the given equation in the coordinate plane is:

Question 11.
Amy began with $25 in her bank account and spent $5 each day. The line shows the amount of money in her bank account. She incorrectly wrote an equation for the line in slope-intercept form as y = -5x + 5.
a. What is the correct equation for the line in slope-intercept form?

Answer:
It is given that
Amy began with $25 in her bank account and spent $5 each day. The line shows the amount of money in her bank account
Now,
The given graph is:

From the given graph,
The y-intercept is: 25
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
The given points to find the slope are: (5, 0), and (1, 20)
So,
Slope (m) = \(\frac{20 – 0}{5 – 1}\)
= \(\frac{20}{5}\)
= 4
So,
The equation of the line in the slope-intercept form is:
y = 4x + 25
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = 4x + 25

b. Critique Reasoning What mistake might Amy have made?
Answer:
Answer:
The mistakes might made by Amy are:
A) The value of y-intercept is 25 and the value of x-intercept is: 5
B) The slope is not negative as it moves down from top to bottom

Question 12.
Higher-Order Thinking The line represents the cost of ordering concert tickets online.
a. Write an equation for the line in slope-intercept form, where x is the number of tickets and y is the total cost.

Answer:
It is given that
The line represents the cost of ordering concert tickets online.
Now,
The given graph is:

From the given graph,
The y-intercept is: 10
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
The given points to find the slope are: (1, 33.25), and (0, 12.25)
So,
Slope (m) = \(\frac{12.25 – 33.25}{0 – 1}\)
= \(\frac{21}{1}\)
= 21
So,
The equation of the line in the slope-intercept form is:
y = 21x + 10
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = 21x + 10

b. Explain how you can write an equation for this situation without using a graph.
Answer:
We know that,
The total cost of ordering concert tickets online = (The cost of 1 Ticket) × (The number of Tickets) + Processing fee
Let the number of tickets be x
Let the total cost of ordering concert tickets online be y
So,
y = 21x + 10
Hence, from the above,
We can conclude that the equation for this situation without using a graph is:
y = 21x + 10

c. Is this graph a good representation of the situation? Explain.
Answer:
Yes,
The given graph is good for the given situation because the equation of the line is the same for this situation with using the graph and without using the graph

Assessment Practice

Question 13.
What should you do first to graph the equation y = \(\frac{2}{5}\)x – 1?
A. Plot the point (0, 0).
B. Plot the point (2, 5).
C. Plot a point at the x-intercept.
D. Plot a point at the y-intercept.
Answer:
The given equation is:
y = \(\frac{2}{5}\)x – 1
Compare the above equation with
y = mx + c
Hence, from the above,
We can conclude that the first step to draw the graph for the given equation is:
Plot a point at the y-intercept

Question 14.
Write an equation for the line in slope-intercept form.

Answer:
The given graph is:

From the given graph,
We can observe that
The y-intercept of the graph is: 8
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
From the given graph,
The points to find the graph are: (4, 0), and (0, 8)
So,
Slope (m) = \(\frac{8 – 0}{0 – 4}\)
= -2
So,
The equation of the line in the slope-intercept form is:
y = mx + c
So,
y = -2x + 8
Hence, from the above,
We can conclude that
The equation of the line in the slope-intercept form is:
y = -2x + 8

TOPIC 2 REVIEW

? Topic Essential Question

How can you analyze connections between linear equations and use them to solve problems?
Answer:
Assuming that your two equations are distinct (neither is merely a multiple of the other), we can use the “elimination by addition and subtraction” method or substitution method to eliminate one variable, leaving us with an equation in one variable, solve this 1-variable (Ex: in x) equation, and then use the resulting value in the other

Vocabulary Review

Complete each definition and provide an example of each vocabulary word.

Question 1.
The change in y divided by the change in x is the ____
Answer:
The change in y divided by the change in x is defined as the “Slope of a line”
Example:
Slope = \(\frac{y}{x}\)
= \(\frac{2}{5}\)

Question 2.
The point on the graph where the line crosses the y-axis is the ____ of a line.
Answer:
The point on the graph where the line crosses the y-axis is the “y-intercept” of a line. In the y-intercept, the value of x is 0
Example:
The point on the graph where the line crosses the y-axis is at (0, 2)
So,
The y-intercept is: 2

Question 3.
The ____ of a line is y = mx + b. The variable m in the equation stands for the __. The variable b in the equation stands for the ___
Answer:
The “Slope-intercept form” of a line is
y = mx + b
The variable m in the equation stands for the x-intercept.
The variable b in the equation stands for the y-intercept

Use Vocabulary in Writing
Paddle boats rent for a fee of $25, plus an additional $12 per hour. What equation, in y = mx + b form, represents the cost to rent a paddle boat for x hours? Explain how you write the equation. Use vocabulary words in your explanation.
Answer:
It is given that
Paddleboats rent for a fee of $25, plus an additional $12 per hour.
Where,
x represents the cost to rent a paddleboat for x hours
Now,
The total cost to rent a paddleboat = The cost of a paddleboat per hour + $12
y = $25x + $12
Hence, from the above,
We can conclude that the equation of the line for this situation is:
y = $25x + $12

Concepts and Skills Review

LESSON 2.1 Combine Like Terms to Solve Equations

Quick Review
You can use variables to represent unknown quantities. To solve an equation, collect like terms to get one variable on one side of the equation. Then use inverse operations and properties of equality to solve the equation.

Practice
Solve each equation for x.

Question 1.
2x + 6x = 1,000
Answer:
The given equation is:
2x + 6x = 1,000
So,
8x = 1,000
Divide by 8 into both sides
x = \(\frac{1,000}{8}\)
x = 125
Hence, from the above,
We can conclude that the value of x is: 125

Question 2.
2\(\frac{1}{4}\)x + 2\(\frac{1}{2}\)x = 44
Answer:
The given equation is:
2\(\frac{1}{4}\)x + 2\(\frac{1}{2}\)x = 44
We know that,
2\(\frac{1}{4}\) = \(\frac{9}{4}\)
2\(\frac{1}{2}\) = \(\frac{5}{2}\)
So,
\(\frac{9}{4}\)x + \(\frac{5}{2}\)x = 44
\(\frac{19}{4}\)x = 44
Multiply with \(\frac{4}{19}\) on both sides
So,
x = 44 × \(\frac{4}{19}\)
x = \(\frac{88}{19}\)
Hence, from the above,
We can conclude that the value of x is: \(\frac{88}{19}\)

Question 3.
-2.3x – 4.2x = -66.3
Answer:
The given equation is:
-2.3x – 4.2x = -66.3
So,
-6.5x = -66.3
6.5x = 66.3
Divide by 6.5 into both sides
So,
x = \(\frac{66.3}{6.5}\)
x = \(\frac{51}{5}\)
x = 10.2
Hence, from the above,
We can conclude that the value of x is: 10.2

Question 4.
Javier bought a microwave for $105. The cost was 30% off the original price. What was the price of the microwave before the sale?
Answer:
It is given that
Javier bought a microwave for $105. The cost was 30% off the original price
So,
The price of the microwave before the sale = The price of the microwave + 30% of the price of the microwave
= $105 + \(\frac{30}{100}\) ($105)
= $105 (\(\frac{130}{100}\))
= \(\frac{13650}{100}\)
= $136.5
Hence, from the above,
We can conclude that the price of the microwave before the sale is: $136.5

LESSON 2.2 Solve Equations with Variables on Both Sides

Quick Review
If two quantities represent equal amounts and have the same variables, you can set the expressions equal to each other. Collect all the variables on one side of the equation and all the constants on the other side. Then use inverse operations and properties of equality to solve the equation.

Practice
Solve each equation for x.

Question 1.
3x + 9x = 6x + 42
Answer:
The given equation is:
3x + 9x = 6x + 42
12x = 6x + 42
Rearrange the like terms
So,
12x – 6x = 42
6x = 42
So,
x = \(\frac{42}{6}\)
x = 7
Hence, from the above,
We can conclude that the value of x is: 7

Question 2.
\(\frac{4}{3}\)x + \(\frac{2}{3}\)x = \(\frac{1}{3}\)x + 5
Answer:
The given equation is:
\(\frac{4}{3}\)x + \(\frac{2}{3}\)x = \(\frac{1}{3}\)x + 5
So,
\(\frac{6}{3}\)x = \(\frac{1}{3}\)x + 5
\(\frac{6}{3}\)x – \(\frac{1}{3}[latex]x = 5
[latex]\frac{5}{3}\)x = 5
Multiply with \(\frac{3}{5}\) on both sides
So,
x = 5 × \(\frac{3}{5}\)
x = 3
Hence, from the above,
We can conclude that the value of x is: 3

Question 3.
9x – 5x + 18 = 2x + 34
Answer:
The given equation is:
9x – 5x + 18 = 2x + 34
So,
4x + 18 = 2x + 34
Rearrange the like terms
So,
4x – 2x = 34 – 18
2x = 16
Divide by 2 into both sides
So,
x = \(\frac{16}{2}\)
x = 8
Hence, from the above,
We can conclude that the value of x is: 8

Question 4.
Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week. After how many weeks will Megan and Connor have saved the same amount?
Answer:
It is given that
Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week.
Now,
Let x be the number of weeks
So,
The money saved by Megan = $50 + $5.50x
The money saved by Connor = $18.50 + $7.75x
So,
To find out after how many weeks Megan and Connor have saved the same amount,
$50 + $5.50x = $18.50 + $7.75x
Rearrange the like terms
So,
$50 – $18.50 = $7.75x – $5.50x
$31.05 = $2.25x
Divide by 2.25 into both sides
So,
x = \(\frac{31.05}{2.25}\)
x = 13.8
x = 14 weeks 1 day
x ≅ 14 weeks
Hence, from the above,
We can conclude that after approximately 14 weeks, Megan and Connor have saved the same amount

LESSON 2.3 Solve Multistep Equations

Quick Review
When solving multistep equations, sometimes the Distributive Property is used before you collect like terms. Sometimes like terms are collected, and then you use the Distributive Property.

Practice Solve each equation for x.

Question 1.
4(x + 4) + 2x = 52
Answer:
The given equation is:
4 (x + 4) + 2x = 52
So,
4 (x) + 4 (4) + 2x = 52
4x + 16 + 2x = 52
6x + 16 = 52
Rearrange the like terms
So,
6x = 52 – 16
6x = 36
x = \(\frac{36}{6}\)
x = 6
Hence, from the above,
We can conclude that the value of x is: 6

Question 2.
8(2x + 3x + 2) = -4x + 148
Answer:
The given equation is:
8 (2x + 3x + 2) = -4x + 148
So,
8 (5x + 2) = -4x + 148
8 (5x) + 8 (2) = -4x + 148
40x + 16 = -4x + 148
Rearrange the like terms
So,
40x + 4x = 148 – 16
44x = 132
x = \(\frac{132}{4}\)
x = 3
Hence, from the above,
We can conclude that the value of x is: 3

Question 3.
Justin bought a calculator and a binder that were both 15% off the original price. The original price of the binder was $6.20. Justin spent a total of $107.27. What was the original price of the calculator?
Answer:
It is given that
Justin bought a calculator and a binder that were both 15% off the original price. The original price of the binder was $6.20. Justin spent a total of $107.27.
So,
Total spent money of Justin = The original price of binder + The original price of a calculator
Let the original price of the calculator be x
So,
$6.20 + 30% of $6.20 + x + 30% of x = $107.27
$6.20 + \(\frac{3}{10}\) ($6.20) + x + \(\frac{3}{10}\) of x = $107.27
$6.20 + 1.86 + 1.3x = $107.27
$8.06 + 1.3x = $107.27
1.3x = $107.27 – $8.06
1.3x = 99.21
x = \(\frac{99.21}{1.3}\)
x = 76.31
Hence, from the above,
We can conclude that the original price of the calculator is: $76.31

LESSON 2.4 Equations with No Solutions or Infinitely Many Solutions

Quick Review
When solving an equation results in a statement that is always true, there are infinitely many solutions. When solving an equation produces a false statement, there are no solutions. When solving an equation gives one value for a variable, there is one solution.

Practice
How many solutions does each equation have?

Question 1.
x + 5.5 + 8 = 5x – 13.5 – 4x
Answer:
The given equation is:
x + 5.5 + 8 = 5x – 13.5 – 4x
So,
x + 13.5 = x – 13.5
Subtract with x on both sides
So,
13.5 = -13.5
Hence, from the above,
we can conclude that there are no solutions for the given equation

Question 2.
4(\(\frac{1}{2}\)x + 3) = 3x + 12 – x
Answer:
The given equation is:
4(\(\frac{1}{2}\)x + 3) = 3x + 12 – x
So,
4 × \(\frac{1}{2}\)x + 4 (3) = 3x + 12 – x
2x + 12 = 2x + 12
Subtract with 2x on both sides
So,
12 = 12
Hence, from the above,
We can conclude that there are infinitely many solutions for the given equation

Question 3.
2(6x + 9 – 3x) = 5x + 21
Answer:
The given equation is:
2 (6x + 9 – 3x) = 5x + 21
So,
2 (3x + 9) = 5x + 21
2 (3x) + 2 (9) = 5x + 21
6x + 18 = 5x + 21
Rearrange the like terms
So,
6x – 5x = 21 – 18
x = 3
Hence, from the above,
We can conclude that there is only 1 solution for the given equation

Question 4.
The weight of Abe’s dog can be found using the expression 2(x + 3), where x is the number of weeks. The weight of Karen’s dog can be found using the expression 3(x + 1), where x is the number of weeks. Will the dogs ever be the same weight? Explain.
Answer:
It is given that
The weight of Abe’s dog can be found using the expression 2(x + 3), where x is the number of weeks. The weight of Karen’s dog can be found using the expression 3(x + 1), where x is the number of weeks.
Now,
To find out whether the weight of the dogs will be the same or not,
2 (2x + 3) = 3 (3x + 1)
So,
2 (2x) + 2 (3) = 3 (3x) + 3 (1)
4x + 6 = 9x + 3
Rearrange the like terms
So,
9x – 4x = 6 – 3
5x = 3
x = \(\frac{3}{5}\)
So,
There is only 1 solution for the given equation
Hence, from the above,
We can conclude that the weights of the dogs will be the same

LESSON 2.5 Compare Proportional Relationships

Quick Review
To compare proportional relationships, compare the rate of change or find the unit rate.

Practice

Question 1.
Two trains are traveling at a constant rate. Find the rate of each train. Which train is traveling at the faster rate?


Answer:
We know that,
Unit rate = \(\frac{y}{x}\)
We know that,
Speed = \(\frac{Distance}{Time}\)
Now,
For Train A,
Unit rate = \(\frac{A value of Distance}{The value of time that corresponds to the Distance}\)
= \(\frac{50}{2}\)
= 25 miles per hour
For Train B,
Unit rate = \(\frac{y}{x}\)
= \(\frac{20}{1}\)
= 20 miles per hour
So,
Unit rate of Train A > Unit rate of Train B
Hence, from the above,
We can conclude that Train A is the fastest

Question 2.
A 16-ounce bottle of water from Store A. costs $1.28. The cost in dollars, y, of a bottle of water from Store B is represented by the equation y = 0.07x, where x is the number of ounces. What is the cost per ounce of water at each store? Which store’s bottle of water costs less per ounce?
Answer:
It is given that
A 16-ounce bottle of water from Store A. costs $1.28. The cost in dollars, y, of a bottle of water from Store B is represented by the equation y = 0.07x, where x is the number of ounces.
So,
The cost per ounce of water of store A = \(\frac{The cost of a 16-ounce bottle of water}{16}\)
= \(\frac{$1.28}{16}\)
= $0.08
The cost per ounce of water of store B = \(\frac{y}{x}\)
= $0.07
So,
The cost per ounce of water of store A > The cost per ounce of water of store B
Hence, from the above,
We can conclude that the cost per ounce of water of store B costs less per ounce

LESSON 2.6 Connect Proportional Relationships and Slope

Quick Review
The slope of a line in a proportional relationship is the same as the unit rate and the constant of proportionality.

Practice

Question 1.
The graph shows the proportions of blue paint and yellow paint that Briana mixes to make green paint. What is the slope of the line? Tell what it means in the problem situation.

Answer:
It is given that
The graph shows the proportions of blue paint and yellow paint that Briana mixes to make green paint.
Now,
The given graph is:

So,
From the graph,
The slope of the given line = \(\frac{y}{x}\)
= \(\frac{5}{6}\)
Hence, from the above slope of the line,
We can conclude that for 5 parts of yellow paint, we have to mix 6 parts of blue paint to make green paint

LESSON 2.7 Analyze Linear Equations: y = mx

Quick Review
A proportional relationship can be represented by an equation in the form y = mx, where m is the slope.

Practice
A mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.

Question 1.
Write a linear equation that represents the relationship between peanuts, x, and walnuts, y.
Answer:
It is given that
A mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.
We know that,
Slope (m) = \(\frac{y}{x}\)
m = \(\frac{1}{3}\)
We know that,
The linear equation that represents the relationship between peanuts and walnuts is:
y = mx
So,
y = \(\frac{1}{3}\)x
x = 3y
Hence, from the above,
We can conclude that the linear equation that represents the relationship between peanuts and walnuts is:
x = 3y

Question 2.
Graph the line.

Answer:
The linear equation that represents the relationship between peanuts and walnuts is:
x = 3y
Hence,
The representation of the linear equation in the coordinate plane is:

LESSON 2.8 Understand the y-Intercept of a Line

Quick Review
The y-intercept is the y-coordinate of the point where a line crosses the y-axis. The y-intercept of a proportional relationship is 0.

Practice
The equation y = 5 +0.5x represents the cost of getting a car wash and using the vacuum for x minutes.

Question 1.
What is the y-intercept?
Answer:
We know that,
The equation of the line in the y-intercept form is:
y = mx + c
Where,
m is the slope
c is the y-intercept
Now,
The given equation is:
y = 5 + 0.5x
Hence, from the above,
We can conclude that the y-intercept is: 5

Question 2.
What does the y-intercept represent?
Answer:
The y-intercept in the given situation represents that the initial cost of getting a car wash using the Vaccum

LESSON 2.9 Analyze Linear Equations: y = mx + b

Quick Review
An equation in the form y = mx + b, where b=0, has a slope of m and a y-intercept of b. This form is called the slope-intercept form. There is not a proportional relationship between x and y in these cases.

Practice

Question 1.
Graph the line with the equation y = \(\frac{1}{2}\)x – 1.

Answer:
The given equation is:
y = \(\frac{1}{2}\)x – 1
Hence,
The representation of the given equation in the coordinate plane is:

Question 2.
What is the equation of the line?

Answer:
The given graph is:

From the given graph,
We can observe that
The y-intercept is: 3
Now,
We know that,
The equation of the line in the slope-intercept form is:
y = mx + c
Now,
To find the slope,
The points are: (0, 3), and (3, 0)
So,
Slope (m) = \(\frac{0 – 3}{3 – 0}\)
= \(\frac{-3}{3}\)
= -1
Hence, from the above,
We can conclude that the equation of the line in the slope-intercept form is:
y = -x + 3

Topic 2 Fluency Practice

Pathfinder

Each block below shows an equation and a possible solution. Shade a path from START to FINISH. Follow the equations that are solved correctly. You can only move up, down, right, or left.