{"id":5426,"date":"2024-01-04T08:30:55","date_gmt":"2024-01-04T03:00:55","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=5426"},"modified":"2024-01-04T09:44:08","modified_gmt":"2024-01-04T04:14:08","slug":"envision-math-common-core-grade-8-answer-key-topic-2","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-8-answer-key-topic-2\/","title":{"rendered":"enVision Math Common Core Grade 8 Answer Key Topic 2 Analyze And Solve Linear Equations"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 8 Answer Key<\/a>\u00a0Topic 2 Analyze And Solve Linear Equations<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 8th Grade Answers Key Topic 2 Analyze And Solve Linear Equations<\/h2>\n

Topic Essential Question<\/strong>
\nHow can we analyze connections between linear equations, and use them to solve problems?
\nAnswer:
\nOne 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.
\nAssuming 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,
\nsolve this 1-variable (x) equation, and then use the resulting value in the other equation to find the value of the other variable (y).
\nBy doing this we find a unique solution (x, y) that satisfies both original equations.
\nNot only that but also this solution (x, y) will also satisfy both of the original linear equations.<\/p>\n

3-ACT MATH<\/strong>
\n\"Envision<\/p>\n

Powering Down<\/strong>
\nDo 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.
\n\"Envision<\/p>\n

Topic 2 enVision STEM Project<\/strong><\/p>\n

Did You Know?<\/strong>
\nDemography is the study of changes, such as the number of births, deaths, or net migration, occurring in the human population over time.
\n\"Envision<\/p>\n

Deaths Worldwide in 2015 (estimated)
\nEmigration 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.
\n\"Envision
\nImmigration is the act of entering and settling in a foreign country. The United States has the largest immigrant population in the world.<\/p>\n

Your Task: Modeling Population Growth<\/strong>
\n\"Envision
\nHuman 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.
\nAnswer:
\nIt 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
\nSo,
\nThe total population at the end of the year = (Total population at the start of the year) – ( Births + Deaths + Immigrants + Emigrants at that year)
\n= 20 million – (600,000 + 350,000 + 100,000 + 5,000)
\n= 20 million – 10.5 million
\n= 9.5 million
\nChange in Population = (Births + Immigration) – (Deaths + Emigration)
\n= (600,000 + 100,000) – (350,000 + 5,000)
\n= 700,000 – 355,000
\n= 345,000
\nHence, from the above,
\nWe can conclude that
\nThe total population at the end of the year is: 9.5 million
\nThe change in population at that year is: 345,000<\/p>\n

\"Analyze<\/p>\n

Topic 2 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong>
\nChoose the best term from the box to complete each definition.
\ninverse operations
\nlike terms
\nproportion
\nvariables<\/p>\n

Question 1.
\nIn an algebraic expression, __________ are terms that have the same variables raised to the same exponents.
\nAnswer:
\nWe know that,
\nIn an algebraic expression, “Like terms” are terms that have the same variables raised to the same exponents.
\nHence, from the above,
\nWe can conclude that the best term that fits the given definition is: Like terms<\/p>\n

Question 2.
\nQuantities that represent an unknown value are _________.
\nAnswer:
\nWe know that,
\nQuantities that represent an unknown value are “Variables”
\nHence, from the above,
\nWe can conclude that the best term that fits the given definition is: Variables<\/p>\n

Question 3.
\nA _________ is a statement that two ratios are equal.
\nAnswer:
\nWe know that,
\nA “Proportion” is a statement that two ratios are equal.
\nHence, from the above,
\nWe can conclude that the best term that fits the given definition is: Proportion<\/p>\n

\"Analyze<\/p>\n

Question 4.
\nOperations that “undo” each other are __________.
\nAnswer:
\nWe know that,
\nOperations that “undo” each other are ” Inverse Operations”
\nHence, from the above,
\nWe can conclude that the best term that fits the given definition is: Inverse Operations<\/p>\n

Identify Like Terms<\/strong><\/p>\n

Complete the statements to identify the like terms in each expression.
\nQuestion 5.
\n4x + 7y – 62 + 6y \u2013 9x
\n4x and ______ are like terms.
\n7y and _______ are like terms.
\nAnswer:
\nThe given expression is:
\n4x + 7y – 62 + 6y – 9x
\nWe know that,
\nThe “Like terms” are terms that have the same variables raised to the same exponents.
\nHence, from the above,
\nWe can conclude that
\n4x and 9x are like terms
\n7y and 6y are like terms<\/p>\n

Question 6.
\n\\(\\frac{1}{2}\\)s \u2013 (6u \u2013 9u) + \\(\\frac{1}{10}\\)s + 25
\n\\(\\frac{1}{2}\\)s and _______ are like terms.
\n6u and _______ are like terms.
\nAnswer:
\nThe given expression is:
\n\\(\\frac{1}{2}\\)s – (6u – 9u) + \\(\\frac{1}{10}\\)s + 25
\n= \\(\\frac{1}{2}\\)s + 9u – 6u + \\(\\frac{1}{10}\\)s + 25
\nWe know that,
\nThe “Like terms” are terms that have the same variables raised to the same exponents.
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{2}\\)s and \\(\\frac{1}{10}\\)s are like terms
\n6u and 9u are like terms<\/p>\n

Solve One-Step Equations<\/strong><\/p>\n

Simplify each equation.
\nQuestion 7.
\n2x = 10
\nAnswer:
\nThe given expression is:
\n2x = 10
\nDivide by 2 into both sides
\n\\(\\frac{2}{2}\\)x = \\(\\frac{10}{2}\\)
\nx = 5
\nHence, from the above,
\nWe can conclude that the value of x is: 5<\/p>\n

Question 8.
\nx + 3 = 12
\nAnswer:
\nThe given expression is:
\nx + 3 = 12
\nSubtract with 3 on both sides
\nx + 3 – 3 = 12 – 3
\nx = 9
\nHence, from the above,
\nWe can conclude that the vaue of x is: 9<\/p>\n

Question 9.
\nx – 7 = 1
\nAnswer:
\nThe given expression is:
\nx – 7 = 1
\nAdd with 7 on both sides
\nx – 7 + 7 = 1 + 7
\nx = 8
\nHence, from the above,
\nWe can conclude that the value of x is: 8<\/p>\n

Simplify Fractions<\/strong><\/p>\n

Question 10.
\nExplain how to simplify the fraction \\(\\frac{12}{36}\\).
\nAnswer:
\nThe given fraction is:
\n\\(\\frac{12}{36}\\)
\nFrom the given fraction,
\nWe can observe that the numerator and the denominator are the multiples of 12
\nSo,
\nDivide the numerator by 12 and the denominator by 12
\nSo,
\n\\(\\frac{12}{36}\\) = \\(\\frac{1}{3}\\)
\nHence,
\nThe simplified form of the given fraction is: \\(\\frac{1}{3}\\)<\/p>\n

Language Development<\/strong>
\nFill in the Venn diagram to compare and contrast linear equations of the form y = mx and y = x + b.
\n\"Envision<\/p>\n

In the box below, draw graphs to represent each form of the linear equations.<\/p>\n

Topic 2 PICK A PROJECT<\/h3>\n

PROJECT 2A<\/strong>
\nIf you had to escape from a locked room, how would you start?
\nPROJECT: DESIGN AN ESCAPE-ROOM ADVENTURE
\n\"Envision<\/p>\n

PROJECT 2B<\/strong>
\nWhat animal would you most like to play with for an hour? Why?
\nPROJECT: PLAN A PET CAF\u00c9
\n\"Envision<\/p>\n

PROJECT 2C<\/strong>
\nIf you wrote a play, what would it be about?
\nPROJECT: WRITE A PLAY
\n\"Envision<\/p>\n

PROJECT 2D<\/strong>
\nHow many tiny steps does it take to cross a slackline?
\nPROJECT: GRAPH A WALKING PATTERN
\n\"Envision<\/p>\n

Lesson 2.1 Combine Like Terms to solve Equations<\/h3>\n

Explore It!<\/strong>
\nA superintendent orders the new laptops shown below for two schools in her district. She receives a bill for $7,500.
\n\"Envision
\nI can… solve equations that have like terms on one side.<\/p>\n

A. Draw a representation to show the relationship between the number of laptops and the total cost.
\nAnswer:
\nIt is given that she receives a bill for $7,500
\nSo,
\nThe total cost of the laptops that are given in the above figure = $7,500
\nNow,
\nLet the cost of a laptop be $x
\nSo,
\n$3x + $4x + $3x = $7,500
\n$10x = $7,500
\nHence, from the above,
\nWe can conclude that
\nThe representation to show the relationship between the number of laptops and the total cost is:
\n$10x = $7,500<\/p>\n

B. Use the representation to write an equation that can be used to determine the cost of one laptop.
\nAnswer:
\nFrom part (a),
\nThe representation to show the relationship between the number of laptops and the total cost is:
\n$10x = $7,500
\nDivide with 10 into both sides
\nSo,
\n\\(\\frac{$10x}{10}\\) = \\(\\frac{$7,500}{10}\\)
\n$x = $750
\nHence, from the above,
\nWe can conclude that
\nThe representation to write an equation that can be used to determine the cost of one laptop is:
\n$x = $750<\/p>\n

\"Analyze<\/p>\n

Focus on math practices<\/strong>
\nReasoning<\/strong> Why is it important to know that each laptop costs the same amount?
\nAnswer:
\nFrom the given figure,
\nWe can observe that all the laptops are of the same type
\nSo,
\nEach laptop will cost the same amount since all the laptops are the same<\/p>\n

Essential Question<\/strong>
\nHow do you solve equations that contain like terms?
\nAnswer:
\nWe will solve the equations that contain like terms by rearranging the like terms on either the left side or the right side<\/p>\n

Try It!<\/strong><\/p>\n

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?
\nSelena buys necklace and bracelet sets for _________ friends.
\n_____ s + ______ s = 53.94
\n______ s = 53.94
\ns = ______
\nAnswer:
\nLet each necklace and each bracelet be s
\nIt is given that
\nThe cost of each necklace is: $9.99
\nThe cost of each bracelet is: $7.99
\nThe total cost of a necklace and a bracelet is: $53.94
\nSo,
\n$9.99s + $7.99s = $53.94
\n$17.98s = $53.94
\n$1798s = $5394
\nDivide by 1798 on both sides
\n\\(\\frac{$1798}{1798}\\)s = \\(\\frac{$5394}{1798}\\)
\ns = 3
\nHence, from the above,
\nWe can conclude that the number of necklace and bracelet sets that Selena buy is: 3<\/p>\n

Convince Me!<\/strong>
\nSuppose the equation is 9.99s + 7.99s + 4.6 = 53.94. Can you combine the s terms and 4.6? Explain.
\nAnswer:
\nThe given equation is:
\n9.99s + 7.99s + 4.6 = 53.94
\nWe know that,
\nWe can only combine the terms only when they are the “Like terms”
\nSo,
\nIn the given equation,
\n9.99s and 7.99s are the like terms
\n53.94 and 4.6 are the like terms
\nHence, from the above,
\nWe can conclude that we can not combine the s terms and 4.6<\/p>\n

\"Analyze<\/p>\n

Try It!<\/strong><\/p>\n

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.
\nAnswer:
\nIt is given that Nat’s grocery bill was $150 which included a 5% club discount
\nNow,
\nLet x be Nat’s bill before the discount
\nSo,
\nTo find Nat’s bill before discount, we have to find the value of 5% of 150 and add its value from 150
\nWe know that,
\nThe value of the bill will always be less after discount when compared to the value of the bill before discount
\nNow,
\nNat’s bill before the discount = (Nat’s bill which included a 5% club discount) + (Value of 5% of 150)
\nx = $150 + (\\(\\frac{5}{100}\\) \u00d7 150)
\nx = $150 + \\(\\frac{5 \u00d7 150}{100}\\)
\nx = $150 + \\(\\frac{750}{100}\\)
\nx = $150 + $7.5
\nx = $157.5
\nHence, from the above,
\nWe can conclude that Nat’s bill before the discount is: $157.5<\/p>\n

Try It!<\/strong><\/p>\n

Solve for d.
\na. –\\(\\frac{1}{4}\\)d – \\(\\frac{2}{5}\\)d = 39
\nAnswer:
\nThe given expression is:
\n–\\(\\frac{1}{4}\\)d – \\(\\frac{2}{5}\\)d = 39
\n-d (\\(\\frac{1}{4}\\) + \\(\\frac{2}{5}\\)) = 39
\n-d (0.25 + 0.40) = 39
\n-d (0.65) = 39
\n-d = \\(\\frac{39}{0.65}\\)
\n-d = \\(\\frac{39 \u00d7 100}{65}\\)
\n-d = 60
\nd = -60
\nHence, from the above,
\nWe can conclude that the value of d is: -60<\/p>\n

b. -9.760 – (-12.81d) = 8.54
\nAnswer:
\nThe given expression is:
\n-9.760 – (-12.81d) = 8.54
\n-9.760 + 12.81d = 8.54
\nRearrange the like terms in the above equation
\nSo,
\n12.81d = 8.54 + 9.760
\n12.81d = 18.3
\nDivide by 12.81 on both sides
\nSo,
\n\\(\\frac{12.81d}{12.81}\\) = \\(\\frac{18.3}{12.81}\\)
\nd = 1.428
\nHence, from the above,
\nWe can conclude that the value of d is: 1.428<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

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.
\n0.8n + 0.6n = 42
\n1.4n = 42 \u2192 Combine like terms.
\n\\(\\frac{1.4 n}{1.4}=\\frac{42}{1.4}\\)
\nn = 30<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question How do you solve equations that contain like terms?
\nAnswer:
\nIn the equations that contain “Like terms”,
\nFirst, 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<\/p>\n

Question 2.
\nLook for Relationships How do you recognize when an equation has like terms?
\nAnswer:
\nWe know that,
\n\u201cLike terms\u201d are terms that have the same variables raised to the same exponents.
\nHence,
\nWhen there are the same variables in the given equation, we can call that terms “Like terms” in the given equation<\/p>\n

Question 3.
\nMake Sense and Persevere in the equation 0.755 – \\(\\frac{5}{8}\\)s = 44, how do you combine the like terms?
\nAnswer:
\nThe given equation is:
\n0.755 – \\(\\frac{5}{8}\\)s = 44
\nWe know that,
\n\u201cLike terms\u201d are terms that have the same variables raised to the same exponents.
\nSo,
\nIn the given equation,
\n0.755 and 44 are the like terms
\nSo,
\n\\(\\frac{5}{8}\\)s = 0.755 + 44
\n\\(\\frac{5}{8}\\)s = 44.755
\nMultiply with \\(\\frac{8}{5}\\) on both sides
\nSo,
\n\\(\\frac{5}{8}\\)s \u00d7 \\(\\frac{8}{5}\\) = 44.755 \u00d7 \\(\\frac{8}{5}\\)
\ns = 71.608
\nHence, from the above,
\nWe can conclude that the value of s is: 71.608<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nHenry is following the recipe card to make a cake. He has 95 cups of flour. How many cakes can Henry make?
\n\"Envision
\nAnswer:
\nIt is given that Henry is following the recipe card to make a cake and he has 95 cups of flour
\nIt is also given that
\nWe need
\n2\\(\\frac{2}{3}\\) cups of flour for the batter
\n\\(\\frac{1}{2}\\) cup of flour for the topping
\nNow,
\nLet the number of cakes be x
\nSo,
\nBy using the flour for the batter and the topping, Henry can make x cakes
\nNow,
\n(2\\(\\frac{2}{3}\\) + \\(\\frac{1}{2}\\))x = 95
\nWe know that,
\n2\\(\\frac{2}{3}\\) = \\(\\frac{8}{3}\\)
\nSo,
\n(\\(\\frac{8}{3}\\) + \\(\\frac{1}{2}\\))x = 95
\n\\(\\frac{19}{6}\\)x = 95
\nMultiply with \\(\\frac{6}{19}\\) on both sides
\nSo,
\n\\(\\frac{19}{6}\\)x \u00d7 \\(\\frac{6}{19}\\) = 95 \u00d7 \\(\\frac{6}{19}\\)
\nx = \\(\\frac{95 \u00d7 6}{19}\\)
\nx = 30
\nHence, from the above,
\nWe can conclude that the number of cakes made by Henry is: 30<\/p>\n

Question 5.
\nA 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?
\nAnswer:
\nIt is given that a city has a population of 350,000 and it has decreased by 30% in the past ten years
\nNow,
\nLet the population of the city ten years ago be: x
\nTo find the population of the city ten years ago,
\nWe have to find the value of 30% of 350,000 and add it to the 350,000
\nThe 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
\nSo,
\nThe population of the city ten years ago = (The population of the city in the past ten years) + (The value of 30% of 350,000)
\nx = 350,000 + \\(\\frac{30}{100}\\) \u00d7 350,000
\nx = 350,000 + \\(\\frac{30 \u00d7 350,000}{100}\\)
\nx = 350,000 + 105,000
\nx = 455,000
\nHence, from the above,
\nWe can conclude that the population of the city ten years ago is: 455,000<\/p>\n

Question 6.
\nSolve the equation \u201312.2z – 13.4z = -179.2.
\nAnswer:
\nThe given equation is:
\n-12.2z – 13.4z = -179.2
\nFrom the given equation,
\nWe can observe that 12.2 and 13.4 are the like terms
\nSo,
\n-z(12.2 + 13.4) = -179.2
\nz(12.2 + 13.4) = 179.2
\nz(25.6) = 179.2
\nDivide by 25.6 into both sides
\nSo,
\n\\(\\frac{25.6}{25.6}\\)z = \\(\\frac{179.2}{25.6}\\)
\nz = 7
\nHence, from the above,
\nWe can conclude that the value of z is: 7<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> In 7 and 8, complete the steps to solve for x.
\nQuestion 7.
\n\\(\\frac{4}{5}\\)x – \\(\\frac{1}{4}\\)x = 11
\n\"Envision
\nAnswer:
\nThe given equation is:
\n\\(\\frac{4}{5}\\)x – \\(\\frac{1}{4}\\)x = 11
\nx (\\(\\frac{4}{5}\\) – \\(\\frac{1}{4}\\)) = 11
\nx (\\(\\frac{16 – 5}{20}\\)) = 11
\n\\(\\frac{11}{20}\\)x = 11
\nMultiply with \\(\\frac{20}{11}\\) on both sides
\nSo,
\n\\(\\frac{20}{11}\\) (\\(\\frac{11}{20}\\)x) = 11 \u00d7 \\(\\frac{20}{11}\\)
\nx = \\(\\frac{11 \u00d7 20}{11}\\)
\nx = 20
\nHence, from the above,
\nWe can conclude that the value of x is: 20<\/p>\n

Question 8.
\n-0.65x + 0.45x = 5.4
\n\"Envision
\nAnswer:
\nThe given equation is:
\n-0.65x + 0.45x = 5.4
\nSo,
\nx (0.45 – 0.65) = 5.4
\nx (-0.20) =5.4
\nDivide by -0.20 into both sides
\nSo,
\n\\(\\frac{-0.20}{-0.20}\\)x = \\(\\frac{5.4}{-0.20}\\)
\nx = -27
\nHence, from the above,
\nWe can conclude that the avlue of x is: -27<\/p>\n

In 9-12, solve for x.
\nQuestion 9.
\n\\(\\frac{4}{9}\\)x + \\(\\frac{1}{5}\\)x = 87
\nAnswer:
\nThe given equation is:
\n\\(\\frac{4}{9}\\)x + \\(\\frac{1}{5}\\)x = 87
\nSo,
\nx (\\(\\frac{4}{9}\\) + \\(\\frac{1}{5}\\)) = 87
\nx (\\(\\frac{20 + 9}{45}\\)) = 87
\n\\(\\frac{29}{45}\\)x = 87
\nMultiply with \\(\\frac{45}{29}\\) on both sides
\nSo,
\n\\(\\frac{45}{29}\\) (\\(\\frac{29}{45}\\)x) = 87 \u00d7 \\(\\frac{45}{29}\\)
\nx = \\(\\frac{87 \u00d7 45}{29}\\)
\nx = 135
\nHence, from the above,
\nWe can conclude that the value of x is: 135<\/p>\n

\"Analyze<\/p>\n

Question 10.
\n-3.8x \u2013 5.9x = 223.1
\nAnswer:
\nThe given equation is:
\n-3.8x – 5.9x = 223.1
\nSo,
\n-x (3.8 + 5.9) = 223.1
\n-x (9.7) =223.1
\nDivide by -9.7 into both sides
\nSo,
\n\\(\\frac{-9.7}{-9.7}\\)x = \\(\\frac{223.1}{-9.7}\\)
\nx = -23
\nHence, from the above,
\nWe can conclude that the avlue of x is: -23<\/p>\n

Question 11.
\nx + 0.15x = 3.45
\nAnswer:
\nThe givene quation is:
\nx + 0.15x = 3.45
\nSo,
\nx (1 + 0.15) = 3.45
\nx (1.15) = 3.45
\nDivide be 1.15 into both sides
\nSo,
\n\\(\\frac{1.15}{1.15}\\)x = \\(\\frac{3.45}{1.15}\\)
\nx = 3
\nHence, from the above,
\nWe can conclude that the value of x is: 3<\/p>\n

Question 12.
\n–\\(\\frac{3}{5}\\)x – \\(\\frac{7}{10}\\) + \\(\\frac{1}{2}\\)x = 56
\nAnswer:
\nThe given equation is:
\n–\\(\\frac{3}{5}\\)x – \\(\\frac{7}{10}\\) + \\(\\frac{1}{2}\\)x = 56
\nx (\\(\\frac{1}{2}\\) – \\(\\frac{3}{5}\\)) – \\(\\frac{7}{10}\\) = 56
\nx (\\(\\frac{5 – 6}{10}\\)) – \\(\\frac{7}{10}\\) = 56
\n–\\(\\frac{1}{10}\\)x = 56 + \\(\\frac{7}{10}\\)
\n–\\(\\frac{1}{10}\\)x = \\(\\frac{560 + 7}{10}\\)
\nMultiply with 10 on both sides
\nSo,
\n–\\(\\frac{10}{10}\\)x = \\(\\frac{567 \u00d7 10}{10}\\)
\n-x = 567
\nx = -567
\nHence, from the above,
\nWe can conclude that the value of x is: -567<\/p>\n

Question 13.
\nA 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.
\nAnswer:
\nIt 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.
\nSo,
\nThe remaining square feet of sheet metal = (Total square feet of sheet metal) – (The total square feet of sheet metal used so far)
\nThe remaining square feet of sheet metal = 8.2 – 2.1
\nThe remaining square feet of sheet metal = 6.1 square feet
\nNow,
\nIt is given that there is$183 worth of sheet metal remaining
\nNow,
\nLet x be the number of sheet metals per square foot
\nSo,
\n6.1x = $183
\nDivide by 6.1 into both sides
\nSo,
\n\\(\\frac{6.1}{6.1}\\)x = \\(\\frac{$183}{6.1}\\)
\nx = 30
\nHence, from the above,
\nWe can conclude that the number of metal sheets per square foot is: 30<\/p>\n

Question 14.
\nMake Sense and Persevere<\/strong> 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?
\nAnswer:
\nIt 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.
\nSo,
\nThe total amount of raisins to make trail mix = (The supply of raisins to make regular mix) + (The supply of raisins to make spicy mix)
\nThe total amount of raisins to make trail mix = \\(\\frac{3}{8}\\) + \\(\\frac{1}{4}\\)
\nThe total amount of raisins to make trail mix = \\(\\frac{5}{8}\\)
\nNow,
\nLet the number of pounds of raisins at the beginning of the week be x
\nSo,
\n\\(\\frac{5}{8}\\)x = 20
\nMultiply with \\(\\frac{8}{5}\\) on both sides
\nSo,
\nx = 20 \u00d7 \\(\\frac{8}{5}\\)
\nx = \\(\\frac{20 \u00d7 8}{5}\\)
\nx = 32 pounds
\nHence, from the above,
\nWe can conclude that the number of pounds of raisins at the beginning of the week is: 32 pounds<\/p>\n

Question 15.
\nMake Sense and Persevere<\/strong> 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?
\n\"Envision
\nAnswer:
\nIt 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
\nNow,
\nLet x be the maximum depth
\nSo,
\n\\(\\frac{1}{6}\\)x + \\(\\frac{2}{3}\\)x = 650
\n\\(\\frac{1 + 4}{6}\\)x = 650
\n\\(\\frac{5}{6}\\)x = 650
\nMultiply with \\(\\frac{6}{5}\\) on both sides
\nSo,
\nx = 650 \u00d7 \\(\\frac{6}{5}\\)
\nx = \\(\\frac{650 \u00d7 6}{5}\\)
\nx = 780 feet
\nHence, from the above,
\nWe can conclude that the maximum depth is: 780 feet<\/p>\n

Question 16.
\nModel with Math<\/strong> Write an equation that can be represented by the bar diagram, then solve.
\n\"Envision
\nAnswer:
\nThe given bar diagram is:
\n\"Envision
\nSo,
\nFrom the bar diagram,
\nThe representation of the equation is:
\n-1.2y + (-4.2y) = -3.78
\n-1.2y – 4.2y = -3.78
\n– (1.2y + 4.2y) = -3.78
\n1.2y + 4.2y = 3.78
\n5.4y = 3.78
\nDivide by 5.4 into both sides
\nSo,
\n\\(\\frac{5.4}{5.4}\\)y = \\(\\frac{3.78}{5.4}\\)
\ny = 0.7
\nHence, from the above,
\nWe can conclude that the value of y is: 0.7<\/p>\n

Question 17.
\nHigher Order Thinking<\/strong> Solve \\(\\frac{2}{3}\\)h – 156 = 3\\(\\frac{13}{24}\\).
\nAnswer:
\nThe given equation is:
\n\\(\\frac{2}{3}\\)h – 156 = 3\\(\\frac{13}{24}\\)
\nWe know that,
\n3\\(\\frac{13}{24}\\) = \\(\\frac{85}{24}\\)
\nSo,
\n\\(\\frac{2}{3}\\)h – 156 = \\(\\frac{85}{24}\\)
\n\\(\\frac{2}{3}\\)h = \\(\\frac{85}{24}\\) + 156
\n0.666h = 3.541 + 156
\n0.666h = 159.541
\nDivide by 0.666 into both sides
\nSo,
\nh = \\(\\frac{159.541}{0.666}\\)
\nh = 239.552
\nHence, from the above,
\nWe can conclude that the value of ‘h’ is: 239.552<\/p>\n

Question 18.
\nModel with Math<\/strong> 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?
\n\"Envision
\nAnswer:
\nIt 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.
\nNow,
\nLet the number of notebooks and binders that Nathan bought be x
\nFrom the figure,
\nIt is given that
\nThe cost of 1 notebook is: $0.95
\nThe cost of 1 binder is: $5.82
\nSo,
\nThe representation of the cost of total notebooks and binders in the form of the equation is:
\n$0.95x + $5.82x = $27.08
\nHence,
\nThe representation of the above equation in the form of a bar diagram is:
\n\"\"<\/p>\n

Assessment Practice<\/strong>
\nQuestion 19.
\nConstruct Arguments<\/strong> 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?
\nA. Added –\\(\\frac{1}{7}\\) to –\\(\\frac{3}{5}\\)
\nB. Subtracted \\(\\frac{1}{7}\\) from –\\(\\frac{3}{5}\\)
\nC. Multiplied 910 by \\(\\frac{26}{35}\\)
\nD. Multiplied 910 by \\(\\frac{35}{26}\\)
\nAnswer:
\nThe given equation is:
\n–\\(\\frac{3}{5}\\)y – \\(\\frac{1}{7}\\)y = 910
\n-y (\\(\\frac{3}{5}\\) + \\(\\frac{1}{7}\\)) = 910
\n–\\(\\frac{26}{35}\\)y = 910
\nMultiply with –\\(\\frac{35}{26}\\) on both sides
\nSo,
\ny = -910 \u00d7 \\(\\frac{35}{26}\\)
\ny = -1,225
\nHence from the above,
\nWe can conclude that the error your friend makes is:
\nMultiplied 910 by \\(\\frac{26}{35}\\)<\/p>\n

Question 20.
\nA 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.
\nPART A<\/strong>
\nDraw a bar diagram to represent the situation.
\nAnswer:
\nIt is given that a 132-inch board is cut into two pieces and one piece is 3 times the length of the other
\nNow,
\nLet the length of 1 piece be x inches
\nSo,
\nThe length of the other piece is: 3x inches
\nSo,
\nThe representation of the given situation in the form of an equation is:
\n3x + x = 132
\nHence,
\nThe representation of the above equation in the form of a bar diagram is:
\n\"\"<\/p>\n

PART B<\/strong>
\nWrite and solve an equation to find the length of the shorter piece.
\nAnswer:
\nFrom part (a),
\nThe equation that represents the given situation is:
\n3x + x = 132
\n4x = 132
\nDivide by 4 into both sides
\nSo,
\nx = \\(\\frac{132}{4}\\)
\nx = 33 inches
\nHence,from the above,
\nWe can conclude that the length of the shorter piece is: 33 inches<\/p>\n

Lesson 2.2 Solve Equations with Variables on Both Sides<\/h3>\n

Solve & Discuss It!<\/strong>
\nJaxson 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.
\n\"Envision
\nI can… solve equations with variables on both sides of the equal sign.
\nAnswer:
\nIt is given that Jaxson and Bryon collected an equal amount of money during the car wash.
\nIt is also given that they collected cash and checks and each check is written for the same amount x
\nSo,
\nThe amount earned by Jaxson = The amount earned by Bryon
\nNow,
\nFrom the given figure,
\nThe amount earned by Jaxson = The amount earned by cash + The amount earned by checks
\n= 15 + 14x
\nThe amount earned by Bryon = The amount earned by cash + The amount earned by checks
\n= 50 + 7x
\nSo,
\nNow,
\n15 + 14x = 50 + 7x
\nSubtract with 7x on both sides
\n15 + 14x – 7x = 50 + 7x – 7x
\n15 + 7x = 50
\nSubtract with 15 on both sides
\n15 + 7x – 15 = 50 – 15
\n7x = 35
\nDivide by 7 on both sides
\n\\(\\frac{7}{7}\\)x = \\(\\frac{35}{7}\\)
\nx = 5
\nSo,
\nThe total amount of money collected by both boys = 15 +14x + 50 + 7x
\n= 21x + 65
\n= 21 (5) + 65
\n= 105 + 65
\n= $170
\nHence, from the above,
\nWe can conclude that the total amount earned by both the boys is: $170<\/p>\n

Reasoning<\/strong>
\nHow can you use an equation to show that expressions are equal?
\nAnswer:
\nCombine 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.
\nIf all of the terms in the two expressions are identical, then the two\u00a0expressions are equivalent.<\/p>\n

Focus on math practices<\/strong>
\nModel with Math<\/strong> 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?
\nAnswer:
\nFrom the given figure,
\nWe can observe that the two boys earned cash and checks
\nSo,
\nThe total amount earned by any boy = The amount earned due to cash + The amount earned due to checks
\nNow,
\nThe amount earned by Jaxson = The amount earned by cash + The amount earned by checks
\n= 15 + 14x
\nThe amount earned by Bryon = The amount earned by cash + The amount earned by checks
\n= 50 + 7x
\nNow,
\nIt is given that the amount earned by both boys are equal
\nSo,
\nThe amount earned by Jaxson = The amount earned by Bryon
\n15 + 14x = 50 + 7x
\nRearrange the like terms
\n14x – 7x = 50 – 15
\n7x = 35
\nHence, from the above,
\nWe can conclude that the representation of the amount collected by each boy in the form of the equation is:
\n7x = 35<\/p>\n

Essential Question<\/strong>
\nHow do you use inverse operations to solve equations with variables on both sides?
\nAnswer:
\nThe “Inverse operations” allow us to undo what has been done to the variable
\nExample:
\nSolve:
\nx<\/span><\/span>+<\/span><\/span>3<\/span><\/span>=<\/span><\/span>8
\nFrom the above equation,
\nWe can observe that\u00a0 <\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0has been added to the variable,\u00a0x<\/span><\/span><\/span><\/span><\/span><\/span><\/span>.
\nWe know that,
\nThe inverse of addition is subtraction
\nSo,
\nBy subtracting 3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>, We can undo the addition.
\nNow,
\nAfter\u00a03<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0was added, the result was equal to\u00a08<\/span><\/span><\/span><\/span><\/span><\/span><\/span>.
\nWe undo the addition, by subtracting\u00a03<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0and see that, the starting amount was\u00a05<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Try It!<\/strong><\/p>\n

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?
\nLet w= the number of weeks.
\n____ w + 8 = _____ w + 10
\n_____ w + 8 = 10
\n_____ w = _____
\nw = _____
\nThe sunflowers are the same height after ________ weeks.
\nAnswer:
\nIt is given that
\nClass 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.
\nNow,
\nLet the number of weeks be w
\nSo,
\nThe height of a sunflower of class A after w weeks = 3\\(\\frac{1}{2}\\)w + 8
\nWe know that,
\n3\\(\\frac{1}{2}\\) = \\(\\frac{7}{2}\\)
\nSo,
\nThe height of a sunflower of class A after w weeks = \\(\\frac{7}{2}\\)w + 8
\nNow,
\nThe height of a sunflower of class B after w weeks = 3\\(\\frac{1}{4}\\)w + 10
\nWe know that,
\n3\\(\\frac{1}{4}\\) = \\(\\frac{13}{4}\\)
\nSo,
\nThe height of a sunflower of class A after w weeks = \\(\\frac{13}{4}\\)w + 10
\nNow,
\nTo make the height of a sunflower from both classes equal,
\nThe height of sunflower of class A after w weeks = The height of sunflower of class B after w weeks
\n\\(\\frac{7}{2}\\)w + 8 = \\(\\frac{13}{4}\\)w + 10
\nRearrange the like terms
\n\\(\\frac{7}{2}\\)w – \\(\\frac{13}{4}\\)w = 10 – 8
\n\\(\\frac{14 – 13}{4}\\)w = 2
\n\\(\\frac{1}{4}\\)w = 2
\nMultiply with 4 on both sides
\n\\(\\frac{4}{4}\\)w = 2 (4)
\nw = 8 weeks
\nHence, from the above,
\nWe can conclude that after 8 weeks, the sunflowers of class A and class B are of the same height<\/p>\n

Convince Me!<\/strong>
\nHow can you check your work to make sure the value of the variable makes the equation true? Explain.
\nAnswer:
\nTo\u00a0make a true equation,\u00a0check your\u00a0math to\u00a0make sure\u00a0that the\u00a0values\u00a0on each side of the equals sign are the same.\u00a0Ensure\u00a0that the numerical\u00a0values\u00a0on both sides of the “=” sign are the same to\u00a0make a true equation.
\nExamples:
\na) 9 = 9 is a true equation
\nb) 5 + 4 = 9 is a true equation<\/p>\n

Try It!<\/strong><\/p>\n

Solve the equation 96 – 4.5y \u2013 3.2y = 5.6y + 42.80.
\nAnswer:
\nThe given equation is:
\n96 – 4.5y – 3.2y = 5.6y + 42.80
\nNow,
\nRearrange the like terms at one side i.e., y-terms to one side and the constant terms to other side
\nSo,
\n-5.6y – 4.5y – 3.2y = 42.80 – 96
\n-13.3y = -53.2
\n13.3y = 53.2
\nDivide by 13.3 into both sides
\nSo,
\n\\(\\frac{13.3}{13.3}\\)y = \\(\\frac{53.2}{13.3}\\)
\ny = 4
\nHence, from the above,
\nWe can conclude that the value of y is: 4<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

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.
\n\"Envision
\n3x + 15 = 4x + 12
\n3x – 3x + 15 = 4x – 3x + 12
\n15 = x + 12
\n15 – 12 = x + 12 – 12
\n3 = x<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How do you use inverse operations to solve equations with variables on both sides?
\nAnswer:
\nThe “Inverse operations” allow us to undo what has been done to the variable
\nExample:
\nSolve:
\nx<\/span><\/span>+<\/span><\/span>3<\/span><\/span>=<\/span><\/span>8
\nFrom the above equation,
\nWe can observe that\u00a0 <\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0has been added to the variable,\u00a0x<\/span><\/span><\/span><\/span><\/span><\/span><\/span>.
\nWe know that,
\nThe inverse of addition is subtraction
\nSo,
\nBy subtracting 3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>, We can undo the addition.
\nNow,
\nAfter\u00a03<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0was added, the result was equal to\u00a08<\/span><\/span><\/span><\/span><\/span><\/span><\/span>.
\nWe undo the addition, by subtracting\u00a03<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0and see that, the starting amount was\u00a05<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Question 2.
\nReasoning<\/strong> Why are inverse operations and properties of equality important when solving equations? Explain.
\nAnswer:
\nAn “Inverse operation” is two operations that undo each other
\nEx: Addition and Subtraction or Multiplication and Division.
\nYou can perform the same inverse operation\u00a0on each side of an equivalent\u00a0equation\u00a0without changing the\u00a0equality.
\nThis gives us a couple of\u00a0properties\u00a0that hold true for all\u00a0equations.<\/p>\n

Question 3.
\nModel with Math<\/strong> 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?
\nAnswer:
\nIt is given that
\nCynthia earns $680 in commissions and is paid $10.25 per hour. Javier earns $410 in commissions and is paid $12.50 per hour.
\nIt is also given that
\nThe representation of the given situation in the form of the equation is:
\n10.25x + 680 = 12.5x + 410
\nFrom the above equation,
\nWe can observe that
\n10.25x is the amount paid to Cynthia per hour and x is the number of hours
\nHence, from the above,
\nWe can conclude that the variable x represents the “Number of hours”<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nMaria 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?
\nAnswer:
\nIt is given that
\nMaria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 plus a 10% commission on his food sales.
\nSo,
\nTo find the number of food sales that will result in Maria and Liam earning the same amount for the week,
\n20%x = $625 + 10%x
\nWhere,
\nx is the number of food sales
\nSo,
\n\\(\\frac{20}{100}\\)x = $625 + \\(\\frac{10}{100}\\)x
\nRearrange the like terms
\n\\(\\frac{20 – 10}{100}\\)x = $625
\n\\(\\frac{10}{100}\\)x = $625
\n\\(\\frac{1}{10}\\)x = $625
\nMultiply with 10 on both sides
\nSo,
\n\\(\\frac{10}{10}\\)x = $625 (10)
\nx = $6,250
\nHence, from the above,
\nWe can conclude that the number of food sales that will make the same amount in the week for Maria and Liam is: $6,250<\/p>\n

Question 5.
\nSelma’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?
\nAnswer:
\nIt is given that
\nSelma’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.
\nNow,
\nLet x be the number of\u00a0 more minutes that each has to work so that they have the same number of boxes
\nSo,
\nTo find the more minutes each need to work in order to have packed the same number of boxes,
\n\\(\\frac{x}{5}\\) + 12 = \\(\\frac{x}{7}\\) + 18
\nRearrange the like terms
\nSo,
\n\\(\\frac{x}{5}\\) – \\(\\frac{x}{7}\\) = 18 – 12
\n\\(\\frac{7x – 5x}{35}\\) = 6
\n\\(\\frac{2x}{35}\\) = 6
\nDivide by 35 into both sides
\nSo,
\n2x = 6 (35)
\nDivide by 2 into both sides
\nSo,
\nx = \\(\\frac{6 (35)}{2}\\)
\nx = 3 (35)
\nx = 105 minutes
\nHence, from the above,
\nWe can conclude that the number of more minutes that each need to work so that the number of boxes becomes equal is: 105 minutes<\/p>\n

Question 6.
\nSolve the equation –\\(\\frac{2}{5}\\)x + 3 = \\(\\frac{2}{3}\\)x + \\(\\frac{1}{3}\\).
\nAnswer:
\nThe given equation is:
\n–\\(\\frac{2}{5}\\)x + 3 = \\(\\frac{2}{3}\\)x + \\(\\frac{1}{3}\\)
\nRearrange the like terms
\nSo,
\n\\(\\frac{2}{3}\\)x + \\(\\frac{2}{5}\\)x = 3 – \\(\\frac{1}{3}\\)
\n\\(\\frac{10 + 6}{15}\\)x = \\(\\frac{9 – 1}{3}\\)
\n\\(\\frac{16}{15}\\)x = \\(\\frac{8}{3}\\)
\nMultiply with \\(\\frac{15}{16}\\) on both sides
\nx = \\(\\frac{8}{3}\\) \u00d7 \\(\\frac{15}{16}\\)
\nx = \\(\\frac{8 \u00d7 15}{3 \u00d7 16}\\)
\nx = \\(\\frac{5}{2}\\)
\nHence, from the above,
\nWe can conclude that the value of x is: \\(\\frac{5}{2}\\)<\/p>\n

Question 7.
\nSolve the equation -2.6b + 4 = 0.9b \u2013 17.
\nAnswer:
\nThe given equation is:
\n-2.6b + 4 = 0.9b – 17
\nRearrange the like terms
\nSo,
\n0.9b + 2.6b = 17 + 4
\n3.5b = 21
\nDivide by 3.5 into both sides
\nSo,
\n\\(\\frac{3.5}{3.5}\\)b = \\(\\frac{21}{3.5}\\)
\nb = 6
\nHence, from the above,
\nWe can conclude that the value of b is: 6<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice In 8 and 9, solve each equation.
\nQuestion 8.
\n6 – 4x = 6x – 8x + 2
\n6 – 4x = ____ + 2
\n6 = _____ + 2
\n____ = _____
\n_______ = x
\nAnswer:
\nThe given equation is:
\n6 – 4x = 6x – 8x + 2
\nSo,
\n6 – 4x = 2 – 2x
\nRearrange the like terms
\nSo,
\n4x – 2x = 6 – 2
\n2x = 4
\nDivide by 2 into both sides
\nSo,
\n\\(\\frac{2}{2}\\)x = \\(\\frac{4}{2}\\)
\nx = 2
\nHence, from the above,
\nWe can conclude that the value of x is: 2<\/p>\n

Question 9.
\n\"Envision
\nAnswer:
\nThe given equation is:
\n\\(\\frac{5}{3}\\)x + \\(\\frac{1}{3}\\)x = 13\\(\\frac{1}{3}\\) + \\(\\frac{8}{3}\\)x
\nRearrange the like terms
\nSo,
\n\\(\\frac{5 + 1}{3}\\)x – \\(\\frac{8}{3}\\)x = 13\\(\\frac{1}{3}\\)
\n\\(\\frac{6 – 8}{3}\\)x = 13\\(\\frac{1}{3}\\)
\n–\\(\\frac{2}{3}\\)x = \\(\\frac{40}{3}\\)
\nMultiply with 3 on both sides
\nSo,
\n-2x = 40
\ndivide by -2 into both sides
\nSo,
\nx = \\(\\frac{-40}{2}\\)
\nx = -20
\nHence, from the above,
\nWe can conclude that the value of x is: -20<\/p>\n

Question 10.
\nTwo 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nTwo 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.
\nNow,
\nLet x be the number of hours
\nSo,
\nTo make the snowfalls of the two towns equal,
\n5 + 3\\(\\frac{1}{2}\\)x = 6 + 2\\(\\frac{1}{4}\\)x
\nWe know that,
\n3\\(\\frac{1}{2}\\) = \\(\\frac{7}{2}\\)
\n2\\(\\frac{1}{4}\\) = \\(\\frac{9}{4}\\)
\nSo,
\n\\(\\frac{7}{2}\\)x – \\(\\frac{9}{4}\\)x = 6 – 5
\n\\(\\frac{14 – 9}{4}\\)x = 1
\n\\(\\frac{5}{4}\\)x = 1
\nMultiply with \\(\\frac{4}{5}\\) on both sides
\nSo,
\nx = \\(\\frac{4}{5}\\)
\nx = 0.8 hours
\nHence, from the above,
\nWe can conclude that after 0.8 hours, the snowfalls of the two towns will be equal<\/p>\n

Question 11.
\nSolve the equation 5.3g + 9 = 2.3g + 15.
\na. Find the value of g.
\nAnswer:
\nThe given equation is:
\n5.3g + 9 = 2.3g + 15
\nRearrange the like terms
\nSo,
\n5.3g – 2.3g = 15 – 9
\n3.0g = 6
\nDivide by 3 into both sides
\n\\(\\frac{3}{3}\\)g = \\(\\frac{6}{3}\\)
\ng = 2
\nHence, from the above,
\nWe can conclude that the value of g is: 2<\/p>\n

b. Explain how you can check that the value \u00b7 you found for g is correct. If your check does not work, does that mean that your result is incorrect? Explain.
\nAnswer:
\nFrom part (a),
\nWe get the value of g : 2
\nSo,
\nWhether the value of g is correct or not, put it in the given equation
\nIf LHS = RHS,
\nThen, your check is correct. Otherwise, your check is not correct
\nNow,
\n5.3g + 9 = 2.3g + 15
\nPut, g = 2
\nSo,
\n5.3 (2) + 9 = 2.3 (2) + 15
\n10.6 + 9 = 4.6 + 15
\n19.6 = 19.6
\nHence, from the above,
\nWe can conclude that the check is correct<\/p>\n

Question 12.
\nSolve the equation 6 – 6x = 5x – 9x – 2.
\nAnswer:
\nThe given equation is:
\n6 – 6x = 5x – 9x – 2
\nSo,
\n6 – 6x = -4x – 2
\nRearrange the like terms
\nSo,
\n-4x + 6x = 6 + 2
\n2x = 8
\nDivide by 2 into both sides
\nSo,
\n\\(\\frac{2}{2}\\)x = \\(\\frac{8}{2}\\)
\nx = 4
\nHence, from the above,
\nWe can conclude that the value of x is: 4<\/p>\n

Question 13.
\nModel with Math<\/strong> 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.
\nAnswer:
\nIt is given that
\nThe 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.
\nNow,
\nLet the population that are moving in and moving out be x
\nWe know that,
\nMoving in will be positive and Moving out will be negative
\nSo,
\nThe population of one town in Florida = 43,425 + 200x – 125x
\nThe population of a nearby town = 45,000 – 150x
\nSo,
\nTo find out after how many months, they will be equal,
\n43,425 + 200x – 125x = 45,000 – 150x
\n43,425 + 75x = 45,000 – 150x
\nRearrange the like terms
\nSo,
\n150x + 75x = 45,000 – 43,425
\n225x = 1,575
\nDivide by 225 into both sides
\n\\(\\frac{225}{225}\\)x = \\(\\frac{1,575}{225}\\)
\nx = 7
\nHence, from the above,
\nWe can conclude that after 7 months, the population of the towns will be equal<\/p>\n

Question 14.
\nVeronica is choosing between two health clubs After how many months will the total cost for each health club be the same?
\n\"Envision
\nAnswer:
\nIt is given that Veronica is choosing between two health clubs
\nNow,
\nLet x be the number of months so that the cost for the two health clubs will be the same
\nNow,
\nThe total health cost of Yoga studio A = 22 + 24.50x
\nThe total health cost of Yoga studio B = 47 + 18.25x
\nSo,
\nTo find out after how many months, the total cost for the two health clubs will be the same,
\n22 + 24.50x = 47 + 18.25x
\nRearrange the like terms
\nSo,
\n47 – 22 = 24.50x – 18.25x
\n25 = 6.25x
\nDivide by 25 into both sides
\nSo,
\n\\(\\frac{25}{25}\\) = \\(\\frac{6.25}{25}\\)x
\n1 = 0.25x
\n\\(\\frac{x}{4}\\) = 1
\nx = 4
\nHence, from the above,
\nWe can conclude that after 4 months, the total cost for the two health clubs will be the same<\/p>\n

Question 15.
\nHigher-Order Thinking<\/strong> 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe 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.
\nNow,
\nLet x be the number of hours
\nSo,
\nThe price of stock A = $12.73 + $0.06x (Since it is increasing)
\nThe price of stock B = $13.48 – $0.14x (Since it is decreasing)
\nNow,
\nTo find out after how many hours, the prices will be the same,
\n$12.73 + $0.06x = $13.48 – $0.14x
\nRearrange the like terms
\nSo,
\n$13.48 – $12.73 = $0.14x + $0.06x
\n$0.75 = $0.2x
\nDivide by 0.2 into both sides
\nSo,
\nx = \\(\\frac{0.75}{0.2}\\)
\nx = 3.75
\nx = 3.60 + 0.15
\nx = 4 hours 15 minutes
\nHence, from the above,
\nWe can conclude that after 4 hours 15 minutes, the prices of the stocks will be equal<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nIn 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.
\nPART A
\n<\/strong>Which equation models the scoring in the final round and the outcome of the contest?
\nA. 12x + 5x – 165 = -12x + 65
\nB. 12x \u2013 5x + 165 = 12x + 65
\nC. 5x \u2013 12x + 165 = 12x + 65
\nD. 12x \u2013 5x \u2013 165 = 12x + 65
\nAnswer:
\nIt is given that
\nIn 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.
\nNow,
\nLet the number of answers be x
\nWe know that,
\nThe points earned for the correct answers will be positive whereas, for the negative answers, they will be negative
\nSo,
\nFor school A,
\nThe number of answers is:
\n12x – 5x = -165
\n12x – 5x + 165 = 0
\nFor school B,
\nThe number of answers is:
\n12x + 0 = -65
\n12x + 65 = 0
\nNow,
\nIt is given that the two schools are tied
\nSo,
\n12x – 5x + 165 = 12x + 65
\nHence, from the above,
\nWe can conclude that option B matches the above-given situation<\/p>\n

PART B<\/strong>
\nHow many answers did each school get correct in the final round?
\nAnswer:
\nFrom part (a),
\nThe equation that models the scoring and outcome of the contest is:
\n12x – 5x + 165 = 12x + 65
\nNow,
\nRearrange the terms
\nSo,
\n12x – 12x – 5x = 65 – 165
\n-5x = -100
\n5x = 100
\nDivide by 5 into both sides
\nSo,
\n\\(\\frac{5}{5}\\)x = \\(\\frac{100}{5}\\)
\nx = 20
\nHence, from the above,
\nWe can conclude that each school gets 20 correct answers in the final round<\/p>\n

Lesson 2.3 Solve Multistep Equations<\/h3>\n

Solve & Discuss It!<\/strong>
\nA 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.
\n\"Envision
\nI can… solve multistep equations and pairs of equations using more than one approach.
\nThere 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.
\nAnswer:
\nIt is given that
\nA 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.
\nNow,
\nThe capacity of each tank = \\(\\frac{1 million}{2}\\) (Since the tank is half-full)
\nWe know that,
\n1 million = 10 lakhs
\nSo,
\nThe capacity of each tank is: 5 Lakh gallons
\nSo,
\nThe capacity of 3 tanks = 5 Lakh gallons (3)
\n= 15 Lakh gallons
\nNow,
\nThe rate of flow of each tank = (The rate of flow of inlet pipes) + (The rate of flow of outlet pipes)
\nWe know that,
\nThe rate of flow for the inlet pipe will be: Positive
\nThe rate of flow for the outlet pipe will be: Negative
\nSo,
\nThe rate of flow of each tank = (45,000 + 25,000) – 60,000
\n= 70,000 – 60,000
\n= 10,000 gallons per hour
\nSince the three pipes are the same, the rate of flow will also be the same
\nSo,
\nThe rate of flow of three tanks = 10,000 (3)
\n= 30,000 gallons per hour
\nNow,
\nIt is given that the number of hours is: x
\nSo,
\nThe number of hours took to fill all the three tanks = \\(\\frac{ The capacity of three tanks } { The rate of flow of the three tanks }\\)
\nx = \\(\\frac{15,00,000}{30,000}\\)
\nx = 50 hours
\nHence, from the above,
\nWe can conclude that the number of hours took to fill the three tanks is: 50 hours<\/p>\n

Reasoning<\/strong>
\nCan you solve the problem in more than one way?
\nAnswer:
\nYes, we can solve the problem in more than one way
\nThe first way:
\nFirst, calculate the capacity and the rate of flow of each tank and multiply both the quantities with 3 since it is for 3 tanks
\nSo,
\nWe will get the time took to fill the three tanks
\nThe second way:
\nCalculate 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<\/p>\n

Focus on math practices<\/strong>
\nUse Structure What are two different ways to simplify the expression 4(3x + 7x + 5) so that it equals 40x – 20? Explain.
\nAnswer:
\nThe given expression is:
\n4 (3x + 7x + 5)
\nA)
\nThe first way:
\nWe know that,
\nThe distributive property is:
\na (b + c) = ab + ac
\nSo,
\n4 (3x + 7x + 5)
\n= 4 (3x) + 4 (7x) + 4 (5)
\n= 12x + 28x + 20
\n= 40x + 20
\nB)
\nThe second way:
\n4 (3x + 7x + 5)
\nFirst, simplify the expression in the brackets
\nSo,
\n4 (3x + 7x + 5)
\n= 4 (10x + 5)
\n= 4 (10x) + 4(5)
\n= 40x + 20<\/p>\n

Essential Question<\/strong>
\nHow can you use the Distributive Property to solve multistep equations?
\nAnswer:
\nLet A, B, and C be the three variables
\nNow,
\nWe know that,
\nThe Distributive Property of multiplication is:
\nA (B + C) = AB + AC
\n(A + B) C = AC + BC
\n(A + C) B = AB + BC<\/p>\n

Try It!<\/strong><\/p>\n

Solve the equation 3(x – 5) \u2013 5x = -25 + 6x.
\n3_____ + 3 \u2219 ______ – 5x = – 25 + 6x
\n_____ – 5x = – 25 + 6x
\n______ x – 15 = – 25 + 6x
\n______ – 15 = -25 + _____ x
\n______ = _____ x ______
\nx = _____ or ______
\nAnswer:
\nThe given equation is:
\n3 (x – 5) – 5x = -25 + 6x
\n3 (x) – 3 (5) – 5x = -25 + 6x
\n3x – 15 – 5x = -25 + 6x
\n-15 – 2x = -25 + 6x
\nRearrange the like terms
\nSo,
\n-15 + 25 = 6x + 2x
\n8x = 10
\nDivide by 8 into both sides
\nx = \\(\\frac{10}{8}\\)
\nx = \\(\\frac{5}{4}\\)
\nHence, from the above,
\nWe can conclude that the value of x is: \\(\\frac{5}{4}\\)<\/p>\n

Convince Me!<\/strong>
\nCan you add x to -5x on the left side of the equation as the first step? Explain.
\nAnswer:
\nNo, we can’t add x to the -5x because from the given equation,
\nWe are getting
\n3x – 5x
\nSo,
\nWe have to add 3x and not x to -5x<\/p>\n

Try It!<\/strong><\/p>\n

Solve the equation -3(-7 \u2013 x) = \\(\\frac{1}{2}\\)(x + 2).
\nAnswer:
\nThe given equation is:
\n-3 (-7 – x) = \\(\\frac{1}{2}\\) (x + 2)
\nSo,
\n-3 [-(x + 7)] = \\(\\frac{1}{2}\\) (x + 2)
\nWe know that,
\n– * – = +
\nSo,
\n3 (x + 7) = \\(\\frac{1}{2}\\) (x + 2)
\nMultiply with 2 on both sides
\nSo,
\n6 (x + 7) = x + 2
\n6 (x) + 6 (7) = x + 2
\n6x + 42 = x + 2
\nRearrange the like terms
\n6x – x = 2 – 42
\n5x = -40
\nDivide by 5 into both sides
\nSo,
\nx = \\(\\frac{-40}{5}\\)
\nx = -8
\nHence, from the above,
\nWe can conclude that the value of x is: -8<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

When solving multistep equations, sometimes you distribute first and then combine like terms.
\n\"Envision
\nSometimes you combine like terms first and then distribute.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How can you use the Distributive Property to solve multistep equations?
\nAnswer:
\nLet A, B, and C be the three variables
\nNow,
\nWe know that,
\nThe Distributive Property of multiplication is:
\nA (B + C) = AB + AC
\n(A + B) C = AC + BC
\n(A + C) B = AB + BC<\/p>\n

Question 2.
\nReasoning<\/strong> What is the first step when solving the equation 3(3x \u2013 5x) + 2 = -8?
\nAnswer:
\nThe given equation is:
\n3 (3x – 5x) + 2 = -8
\nUse the distributive property of multiplication
\nSo,
\n3 (3x) – 3(5x) + 2 = -8 ——–> First step when solving the above equation<\/p>\n

Question 3.
\nUse Structure<\/strong> 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?
\nAnswer:
\nThe given equation is:
\n7 (x + 5) – x = 42
\nTo find the order of operations, We have to use the BODMAS rule
\nSo,
\nFrom the above equation,
\nWe will first solve the expression present in the brackets, then add, and then subtract
\nWe know that,
\nWe can do any operation only on the like terms
\nWe know that,
\nThe “Like terms” are the terms that have the same exponent
\nSo,
\nFor the above equation,
\nWe can not combine the terms before using the distributive property
\nNow,
\n7 (x) + 7 (5) – x = 42
\n7x + 35 – x = 42
\n6x + 35 = 42
\nSubtract with 35 on both sides
\n6x = 42 – 35
\n6x = 7
\nDivide by 6 into both sides
\nSo,
\nx = \\(\\frac{7}{6}\\)
\nHence, from the above,
\nWe can conclude that the value of x is: \\(\\frac{7}{6}\\)<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nSolve the equation 3x + 2 = x + 4(x + 2).
\nAnswer:
\nThe given equation is:
\n3x + 2 = x + 4 (x + 2)
\n3x + 2 = x + 4 (x) + 4 (2)
\n3x + 2 = x + 4x + 8
\n3x + 2 = 5x + 8
\nRearrange the like terms
\nSo,
\n5x – 3x = -8 + 2
\n2x = -6
\nDivide by 2 into both sides
\nSo,
\nx = \\(\\frac{-6}{2}\\)
\nx = -3
\nHence, from the above,
\nWe can conclude that the value of x is: -3<\/p>\n

Question 5.
\nSolve the equation -3(x – 1) + 7x = 27.
\nAnswer:
\nThe given equation is:
\n-3 (x – 1) + 7x = 27
\nSo,
\n-3 (x) + 3 (1) + 7x = 27
\n-3x + 3 + 7x = 27
\n4x + 3 = 27
\nSubtract with 3 on both sides
\nSo,
\n4x = 27 – 3
\n4x = 24
\nDivide by 4 into both sides
\nSo,
\nx = \\(\\frac{24}{4}\\)
\nx = 6
\nHence, from the above,
\nWe can conclude that the value of x is: 6<\/p>\n

Question 6.
\nSolve the equation \\(\\frac{1}{3}\\)(x + 6) = \\(\\frac{1}{2}\\)(x – 3).
\nAnswer:
\nThe given equation is:
\n\\(\\frac{1}{3}\\)(x + 6) = \\(\\frac{1}{2}\\)(x – 3)
\nMultiply 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)
\nSo,
\n\\(\\frac{6}{3}\\) (x + 6) = \\(\\frac{6}{2}\\) (x – 3)
\n2 (x + 6) = 3 (x – 3)
\n2 (x) + 2 (6) = 3 (x) – 3 (3)
\n2x + 12 = 3x – 9
\nRearrange the like terms
\nSo,
\n3x – 2x = 12 + 9
\nx = 21
\nHence,f rom the above,
\nWe can conclude that the value of x is: 21<\/p>\n

Question 7.
\nSolve the equation 0.25(x + 4) \u2013 3 = 28.
\nAnswer:
\nThe given equation is:
\n0.25 (x + 4) – 3 = 28
\nAdd with 3 on both sides
\nSo,
\n0.25 (x + 4) = 28 + 3
\n0.25 (x + 4) = 31
\nWe know that,
\n0.25 = \\(\\frac{1}{4}\\)
\nSo,
\n\\(\\frac{x + 4}{4}\\) = 31
\nMultiply with 4 on both sides
\nSo,
\nx + 4 = 31 (4)
\nx + 4 = 124
\nSubtract with 4 on both sides
\nSo,
\nx = 124 – 4
\nx = 120
\nHence, from the above,
\nWe can conclude that the value of x is: 120<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> In 8-10, find the value of x.
\nQuestion 8.
\nLori 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?
\n\"Envision
\nThe original price of the sunglasses was _________.
\nAnswer:
\nIt is given that
\nLori bought sunglasses and flip-flops at a half-off sale. If she spent a total of $21 on the two items
\nNow,
\nLet x be the price of sunglasses
\nIt is also given that
\nthe price of flipflops is: $24
\nSo,
\n\\(\\frac{1}{2}\\) (x + $24) = $21
\nMultiply with 2 on both sides
\nSo,
\nx + $24 = $21 (2)
\nx + $24 = $42
\nSubtract with $24 on both sides
\nSo,
\nx = $42 – $24
\nx = $18
\nHence, from the above,
\nWe can conclude that the price of sunglasses is: $18<\/p>\n

Question 9.
\nUse the Distributive Property to solve the equation 28 \u2013 (3x + 4) = 2(x + 6) + x.
\n28 – ______ x – _____ = 2x + _____ + x
\n24 – _____x = ______x + ______
\n24 – _____x = ______
\n_____ x = ______
\nx = ______
\nAnswer:
\nThe given equation is:
\n28 – (3x + 4) = 2 (x + 6) + x
\nBy using the distributive property,
\n28 – 3x – 4\u00a0 = 2 (x) + 2 (6) + x
\n24 – 3x = 2x + 12 + x
\n24 – 3x = 3x + 12
\nRearrange the like terms
\nSo,
\n3x + 3x = 24 – 12
\n6x = 12
\nDivide by 6 into both sides
\nSo,
\nx = \\(\\frac{12}{6}\\)
\nx = 2
\nHence, from the above,
\nWe can conclude that the value of x is: 2<\/p>\n

Question 10.
\nUse the Distributive Property to solve the equation 3(x – 6) + 6 = 5x – 6.
\nx – _____ + 6 = 5x – ______
\n_____ x – _____ = 5x – _______
\n_____ x – _____ = _______
\n______ x = _______
\nx = ________
\nAnswer:
\nThe given equation is:
\n3 (x – 6) + 6 = 5x – 6
\nBy using the Distributive property,
\n3 (x) – 3 (6) + 6 = 5x – 6
\n3x – 18 + 6 = 5x – 6
\n3x – 12 = 5x – 6
\nRearrange the like terms
\nSo,
\n5x – 3x = 6 – 12
\n2x = -6
\nx = \\(\\frac{-6}{2}\\)
\nx = -3
\nHence, from the above,
\nWe can conclude that the value of x is: -3<\/p>\n

Question 11.
\nWhat is the solution to -2.5(4x \u2013 4) = -6?
\nAnswer:
\nThe given equation is:
\n-2.5 (4x – 4) = -6
\nSo,
\n-2.5 (4x) + 2.5 (4) = -6
\n-10x + 10 = -6
\nSubtract with 10 on both sides
\nSo,
\n-10x = -6 – 10
\n-10x = -16
\n10x = 16
\nDivide by 10 into both sides
\nSo,
\nx = \\(\\frac{16}{10}\\)
\nx = 1.6
\nHence, from the above,
\nWe can conclude that the solution of the given equation is: 1.6<\/p>\n

Question 12.
\nWhat is the solution to the equation 3(x + 2) = 2(x + 5)?
\nAnswer:
\nThe given equation is:
\n3 (x + 2) = 2 (x + 5)
\nSo,
\n3 (x) + 3 (2) = 2 (x) + 2 (5)
\n3x + 6 = 2x + 10
\nRearrange the like terms
\nSo,
\n3x – 2x = 10 – 6
\nx = 4
\nHence, from the above,
\nWe can conclude that the solution of the given equation is: 4<\/p>\n

Question 13.
\nSolve the equation \\(\\frac{1}{6}\\)(x \u2013 5) = \\(\\frac{1}{2}\\)(x + 6).
\nAnswer:
\nThe given equation is:
\n\\(\\frac{1}{6}\\)(x \u2013 5) = \\(\\frac{1}{2}\\)(x + 6)
\nMultiply with 6 on both sides
\nSo,
\nx – 5 = 3 (x + 6)
\nx – 5 = 3 (x) + 3 (6)
\nx – 5 = 3x + 18
\nRearrange the like terms
\nSo,
\nx – 3x = 18 + 5
\n-2x = 23
\nDivide by -2 into both sides
\nSo,
\nx = –\\(\\frac{23}{2}\\)
\nHence, from the above,
\nWe can conclude that the value of x for the given equation is: –\\(\\frac{23}{2}\\)<\/p>\n

Question 14.
\nSolve the equation 0.6(x + 2) = 0.55(2x + 3).
\nAnswer:
\nThe given equation is:
\n0.6 (x + 2) = 0.55 (2x + 3)
\nSo,
\n0.6 (x) + 0.6 (2) = 0.55 (2x) + 0.55 (3)
\n0.6x + 1.2 = 1.10x + 1.65
\nRearrange the like terms
\nSo,
\n1.10x – 0.6x = 1.2 – 1.65
\n0.5x = -0.45
\nDivide by 0.5 into both sides
\nSo,
\nx = \\(\\frac{-0.45}{0.5}\\)
\nx = -0.9
\nHence, from the above,
\nWe can conclud ethat the value of x is: -0.9<\/p>\n

Question 15.
\nSolve the equation 4x – 2(x – 2) = -9 + 5x – 8.
\nAnswer:
\nThe given equation is:
\n4x – 2 (x – 2) = -9 + 5x – 8
\nSo,
\n4x – 2 (x) + 2 (2) = -9 + 5x – 8
\n4x – 2x + 4 = 5x – 17
\n2x + 4 = 5x – 17
\nRearrange the like terms
\nSo,
\n5x – 2x = 17 + 4
\n3x = 21
\nDivide by 3 into both sides
\nSo,
\nx = \\(\\frac{21}{3}\\)
\nx = 7
\nHence, from the above,
\nWe can conclude that the value of x is: 7<\/p>\n

Question 16.
\nUse 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.
\nAnswer:
\nThe given equation is:
\n2 (m + 2) = 22
\nWe know that,
\nBy using the distributive property of multiplication,
\nA (B + c) = AB + AC
\nSo,
\n2 (m) + 2 (2) = 22
\n2m + 4 = 22
\n2m = 22 – 4
\n2m = 18
\nm = \\(\\frac{18}{2}\\)
\nm = 9
\nHence, from the above,
\nWe can conclude that the value of m is: 9<\/p>\n

Question 17.
\nWhat is Peter’s number?
\n\"Envision
\nAnswer:
\nLet peter’s number be x
\nSo,
\nAccording to the given statement,
\n-3 (x – 12) = -54
\n3 (x – 12) = 54
\n3x – 3 (12) = 54
\n3x – 36 = 54
\n3x = 36 + 54
\n3x = 90
\nx = \\(\\frac{90}{3}\\)
\nx = 30
\nHence, from the above,
\nWe acn conclude that peter’s number is: 30<\/p>\n

Question 18.
\nHigher Order Thinking<\/strong> Use the Distributive Property to solve the equation \\(\\frac{4x}{5}\\) – x = \\(\\frac{x}{10}\\) – \\(\\frac{9}{2}\\)
\nAnswer:
\nThe given equation is:
\n\\(\\frac{4x}{5}\\) – x = \\(\\frac{x}{10}\\) – \\(\\frac{9}{2}\\)
\nRearrange the like terms
\nSo,
\n\\(\\frac{4x}{5}\\) – x – \\(\\frac{x}{10}\\) = –\\(\\frac{9}{2}\\)
\n\\(\\frac{7x}{10}\\) – x = –\\(\\frac{9}{2}\\)
\n–\\(\\frac{3x}{10}\\) = –\\(\\frac{9}{2}\\)
\nMultiplywith \\(\\frac{10}{3}\\) on both sides
\nx = \\(\\frac{9 \u00d7 10}{2 \u00d7 3}\\)
\nx = 15
\nHence, from the above,
\nWe can conclude that the value of x is: 15<\/p>\n

Assessment Practice<\/strong>
\nQuestion 19.
\nHow many solutions does the equation -2(x + 4) = -2(x + 4) – 6 have?
\nAnswer:
\nThe given equation is:
\n-2 (x + 4) = -2 (x + 4) – 6
\nSo,
\n-2 (x) – 2 (4) = -2 (x) – 2 (4) – 6
\n-2x – 8 = -2x – 8 – 6
\nRearrange the like terms
\nSo,
\n-2x + 2x – 8 + 8 = -6
\n0 = -6
\nHence, from the above,
\nWe can conclude that there are no solutions for the given equation<\/p>\n

Question 20.
\nSolve the equation 3(x + 4) = 2x + 4x \u2013 6 for x.
\nAnswer:
\nThe given equation is:
\n3 (x + 4) = 2x + 4x – 6
\nSo,
\n3 (x) + 3 (4) = 6x – 6
\n3x + 12 = 6x – 6
\nRearrange the like terms
\nSo,
\n6x – 3x = 12 + 6
\n3x = 18
\nx = \\(\\frac{18}{3}\\)
\nx = 6
\nHence, from the above,
\nWe can conclude that the solution of the given equation is: 6<\/p>\n

Lesson 2.4 Equations with No Solutions or Infinitely Many Solutions<\/h3>\n

Explore It!<\/strong>
\nThe 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!
\n\"Envision
\nI can… determine the number of solutions an equation has.<\/p>\n

A. Write expressions to represent Great Karlo’s instructions to each twin.
\nAnswer:
\nIt is given that
\nThe 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.
\nNow,
\nGreat Karlo’s instructions to Jasmine:
\nLet the age of Jasmine be x
\nStep 1:
\nMultiply your age by 3 and add 6
\n3x + 6
\nStep 2:
\nMultiply step 1 with 2
\n2 (3x +6)
\nSo,
\nThe expression representing the age of Jasmine is: 2 (3x + 6)
\nGreat Karlo’s instructions to James
\nLet the age of James be x
\nStep 1:
\nMultiply your age by 2 and add 4
\n2x + 4
\nStep 2:
\nMultiply step 1 with 3
\n3 (2x +4)
\nSo,
\nThe expression representing the age of James is: 3 (2x + 4)
\nHence, from the above,
\nWe can conclude that the expressions that represent the Great Karlo’s instruction to each twin are:
\nFor Jasmine —–> 2 (3x + 6)
\nFor James ——-> 3 (2x + 4)<\/p>\n

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.
\nAnswer:
\nIt is given that the great Karlo predicted that the twins will get the same number
\nSo,
\n2 (3x + 6) = 3 (2x + 4)
\n2 (3x) + 2 (6) = 3 (2x) + 3 (4)
\n6x + 12 = 6x + 12
\nHence,
\nThe table to show the numbers tried for Jasmine’s and James ages and the results are:
\n\"\"<\/p>\n

C. What do you notice about your results?
\nAnswer:
\nFrom the table that is present in part (b),
\nWe can observe that the ages of Jasmine and James are the same<\/p>\n

Focus on math practices<\/strong>
\nMake Sense and Persevere<\/strong> 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.
\nAnswer:
\nThe table that represents three more values of Jasmine’s and James’ ages and its results are:
\n\"\"
\nHence, from the above table,
\nWe can observe that the ages of Jasmine and James are the same
\nHence, from the above,
\nWe can conclude that for any type of the whole number, the ages of Jasmine and James are the same<\/p>\n

Essential Question<\/strong>
\nWill a one-variable equation always have only one solution?
\nAnswer:
\nEvery linear\u00a0equation\u00a0that is a conditional\u00a0equation has one solution. However, not every linear\u00a0equation\u00a0in\u00a0one variable has a single solution. There\u00a0are\u00a0two other cases: no\u00a0solution\u00a0and the\u00a0solution\u00a0set of all real numbers<\/p>\n

Try It!<\/strong><\/p>\n

How many solutions does the equation
\n3x + 15 = 2x + 10 + x + 5 have?
\nThe equation has ______ solutions.
\n3x + 15 = 2x + 10 + x + 5
\n3x + 15 = _____ x + ______
\n3x – _____ + 15 = 3x – _____ + 15
\n______ = _______
\nAnswer:
\nThe given equation is:
\n3x + 15 = 2x + 10 + x + 5
\nSo,
\n3x + 15 = 3x + 15
\nSubtract with 3x on both sides
\nSo,
\n15 = 15
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

Convince Me!<\/strong>
\nIf the value of x is negative, would the equation still be true? Explain.
\nAnswer:
\nFor the given equation,
\n3x + 15 = 2x + 10 + x + 5,
\nThe 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
\nHence, from the above,
\nWe can conclude that the given equation would still be true even if the value of x is negative<\/p>\n

Try It!<\/strong><\/p>\n

How many solutions does the equation 4x + 8 = 0.1x + 3 + 3.9x have? Explain.
\nAnswer:
\nThe given equation is:
\n4x + 8 = 0.1x + 3 + 3.9x
\nSo,
\n4x + 8 = 4x + 3
\nSubtract with 4x on both sides
\nSo,
\n8 = 3
\nHence, from the above,
\nWe can conclude that the given equation has no solutions<\/p>\n

Try It!<\/strong><\/p>\n

Determine the number of solutions each equation has without solving. Explain your reasoning.
\na. 3x + 1.5 = 2.5x + 4.7
\nAnswer:
\nThe give equation is:
\n3x + 1.5 = 2.5x + 4.7
\nRearrange the like terms
\nSo,
\n3x – 2.5x = 4.7 – 1.5
\n0.5x = 3.2
\nDivide by 0.5 into both sides
\nSo,
\nx = \\(\\frac{3.2}{0.5}\\)
\nx = 6.4
\nHence, from the above,
\nWe can conclude that the given equation ahs only 1 solution<\/p>\n

b. 3(x + 2) = 3x – 6
\nAnswer:
\nThe given equation is:
\n3 (x + 2) = 3x – 6
\nSo,
\n3 (x) + 3 (2) = 3x – 6
\n3x + 6 = 3x – 6
\nSubtract with 3x on both sides
\nSo,
\n6 = -6
\nHence, from the above,
\nWe can conclude that the given equation has no solutions<\/p>\n

c. 9x – 4 = 5x \u2013 4 + 4x
\nAnswer:
\nThe given equation is:
\n9x – 4 = 5x – 4 + 4x
\nSo,
\n9x – 4 = 9x – 4
\nSubtract with 9x on both sides
\nSo,
\n-4 = -4
\n4 = 4
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A one-variable equation has infinitely many solutions when solving results in a true statement, such as 2 = 2.
\nA one-variable equation has one solution when solving results in one value for the variable, such as x = 2.
\nA one-variable equation has no solution when solving results in an untrue statement, such as 2 = 3.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> Will a one-variable equation always have only one solution?
\nAnswer:
\nEvery linear\u00a0equation\u00a0that is a conditional\u00a0equation has one solution. However, not every linear\u00a0equation\u00a0in\u00a0one variable has a single solution. There\u00a0are\u00a0two other cases: no\u00a0solution\u00a0and the\u00a0solution\u00a0set of all real numbers<\/p>\n

Question 2.
\nUse Structure<\/strong> Kaylee writes the equation 6x + 12 = 2(3x + 6). Can you find the number of solutions this equation has without solving for x? Explain.
\nAnswer:
\nThe given equation is:
\n6x + 12 = 2 (3x + 6)
\nSo,
\n6x + 2 = 2 (3x) + 2 (6)
\n6x + 12 = 6x + 12
\nSubtract with 12 on both sides
\nSo,
\n12 = 12
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> 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.
\nAnswer:
\nIt is given that
\nThe 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)
\nSo,
\nNow,
\nTo find out whether the two plants will ever be the same height or not,
\n3 (4x + 2) = 6 (2x + 2)
\nSo,
\n3 (4x) + 3 (2) = 6 (2x) + 6 (2)
\n12x + 6 = 12x + 12
\nSubtract with 12x on both sides
\nSo,
\n6 = 12
\nSo,
\nThe given equation has no solution
\nHence, from the above,
\nWe can conclude that it is not possible the two plants will ever be the same height<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nHow many solutions does the equation 3(2.4x + 4) = 4.1x + 7 + 3.1x have? Explain.
\nAnswer:
\nThe given equation is:
\n3 (2.4x + 4) = 4.1x + 7 + 3.1x
\nSo,
\n3 (2.4x) + 3 (4) = 7.2x + 7
\n7.2x + 12 = 7.2x + 7
\nSubtract with 7.2x on both sides
\nSo,
\n12 = 7
\nHence, from the above,
\nWe can conclude that the given equation has no solutions<\/p>\n

Question 5.
\nHow many solutions does the equation 7x + 3x – 8 = 2(5x – 4) have? Explain.
\nAnswer:
\nThe given equation is:
\n7x + 3x – 8 = 2 (5x – 4)
\nSo,
\n10x – 8 = 2 (5x) – 2 (4)
\n10x – 8 = 10x – 8
\nSubtract with 10x on both sides
\nSo,
\n-8 = -8
\n8 = 8
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

Question 6.
\nTodd 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nTodd 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.
\nNow,
\nLet the number of pieces of fruit be x
\nSo,
\nThe amount purchased by Todd = $1.25x + $4
\nThe amount purchased by Agnes = $1x + $6
\nNow,
\nTo find whether they pay the same amount for purchase or not,
\n$1.25x + $4 = $1x + $6
\nRearrange the like terms
\nSo,
\n$1.25x – $1x = $6 – $4
\n$0.25x = $2
\nDivide by 0.25 into both sides
\nSo,
\nx = \\(\\frac{2}{0.25}\\)
\nx = 8
\nHence, from the above,
\nWe can conclude that if there are 8 fruits, then Todd and Agnes will pay the same amount for purchase<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> In 7 and 8, complete the equations to find the number of solutions.
\nQuestion 7.
\nClassify the equation 33x + 99 = 33x \u2013 99 as having one solution, no solution, or infinitely many solutions.
\n33x + 99 = 33x – 99
\n33x – ______ + 99 = 33x – _____ – 99
\n99 ______ – 99
\nSince 99 is _______ equal to -99, the equation has _______ solution(s).
\nAnswer:
\nThe given equation is:
\n33x + 99 = 33x – 99
\nSubtract with 33x on both sides
\nSo,
\n33x – 33x + 99 = 33x – 33x – 99
\n99 = -99
\nWe know that,
\n99 \u2260 -99
\nHence, from the above,
\nWe can conclude that there are no solutions for the given equation<\/p>\n

Question 8.
\nSolve 4(4x + 3) = 19x + 9 – 3x + 3. Does the equation have one solution, no solution, or infinitely many solutions?
\n4(4x + 3) = 19x + 9 – 3x + 3
\n4 \u2022 ______ + 4 \u2022 ______ = 19x + 9 – 3x + 3
\n16x + 12 = _______ + _______
\n16x – ______ + 12 = 16x ______ + 12
\n12 _______ 12
\nSince 12 is ________ equal to 12, the equation has ________ solution(s).
\nAnswer:
\nThe given equation is:
\n4 (4x + 3) = 19x + 9 – 3x + 3
\nSo,
\n4 (4x) + 4 (3) = 16x + 12
\n16x + 12 = 16x + 12
\nSubtract with 16x on both sides
\nSo,
\n16x – 16x + 12 = 16x – 16x + 12
\n12 = 12
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

Question 9.
\nGeneralize<\/strong> What does it mean if an equation is equivalent to 0 = 0? Explain.
\nAnswer:
\nIf an equation is equivalent to 0 = 0, then
\nThe equation is true for all the values of x
\nHence,
\nThat equation has infinitely many solutions<\/p>\n

Question 10.
\nSolve 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.
\nAnswer:
\nThe given equation is:
\n4x + x + 4 = 8x – 3x + 4
\nSo,
\n5x + 4 = 5x + 4
\nSubtract with x on both sides
\nSo,
\n5x – 5x + 4 = 5x – 5x + 4
\n4 = 4
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

Question 11.
\nReasoning<\/strong> 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 \u2013 2. Store B’s prices are represented by the expression 3(5x + 7). When do the two stores charge the same rate? Explain.
\nAnswer:
\nIt\u00a0 is given that
\nTwo 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 \u2013 2. Store B’s prices are represented by the expression 3(5x + 7)
\nSo,
\nTo find when the two stores charge the same rate,
\n15x – 2 = 3 (5x + 7)
\nSo,
\n15x – 2 = 3 (5x) + 3 (7)
\n15x – 2 = 15x – 21
\nSubtract with 15x on both sides
\nSo,
\n15x – 15x – 2 = 15x – 15x – 21
\n-2 = -21
\n2 = 21
\nSo,
\nThe equation has no solution
\nHence, from the above,
\nWe can conclude that the two stores will never charge the same rate<\/p>\n

Question 12.
\nReasoning<\/strong> How is solving an equation with no solution similar to solving an equation that has an infinite number of solutions?
\nAnswer:
\nNo solution\u00a0would mean that there is\u00a0no\u00a0answer to the\u00a0equation. It is impossible for the\u00a0equation\u00a0to be true\u00a0no\u00a0matter what value we assign to the variable.\u00a0Infinite solutions\u00a0would mean that any value for the variable would make the\u00a0equation\u00a0true.<\/p>\n

Question 13.
\nSolve 0.9x + 5.1x – 7 = 2(2.5x \u2013 3). How many solutions does the equation have?
\nAnswer:
\nThe given equation is:
\n0.9x + 5.1x – 7 = 2 (2.5x – 3)
\nSo,
\n6.0x – 7 = 2 (2.5x) – 2 (3)
\n6x – 7 = 5x – 6
\nRearrange the like terms
\nSo,
\n6x – 5x = 7 – 6
\nx = 1
\nHence, from the above,
\nWe can conclude that the given equation has only 1 solution<\/p>\n

Question 14.
\nCritique Reasoning<\/strong> Your friend solved the equation 4x + 12x \u2013 6 = 4(4x + 7) and got x = 34.
\nWhat error did your friend make? What is the correct solution?
\n4x + 12x – 6 = 4 (4x + 7).
\n16x – 6 = 16x + 28
\n16x – 16x – 6 = 16x – 16x + 28
\nx – 6= 28
\nx – 6 + 6 = 28 + 6
\nx = 34
\nAnswer:
\nThe given equation is:
\n4x + 12x – 6 = 4 (4x + 7)
\nSo,
\n16x – 6 = 4 (4x) + 4 (7)
\n16x – 6 = 16x + 28
\nSubtract with 16x on both sides
\nSo,
\n16x – 16x – 6 = 16x – 16x + 28
\n-6 = 28
\nSo,
\nFrom the above,
\nWe can observe that after subtracting the given equation with 16x, there are no x terms.
\nSo,
\nWe 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
\nHence, from the above,
\nWe can conclude that the correct solution for the given equation is: No solutions for the given equation<\/p>\n

Question 15.
\nSolve 49x + 9 = 49x + 83.
\na. Does the equation have one solution, no solution, or infinitely many solutions?
\nAnswer:
\nThe given equation is:
\n49x + 9 = 49x + 83
\nSubtract with 49x on both sides
\nSo,
\n49x – 49x + 9 = 49x – 49x + 83
\n9 = 83
\nHence, from the above,
\nWe can conclude that the given equation has no solutions<\/p>\n

b. Write two equations in one variable that have the same number of solutions as this equation.
\nAnswer:
\nThe two equations in one variable that have the same number of solutions as the equation that is present in part (a) are:
\nA) 10x + 8 = 10x – 25
\nB) 5 (3x + 10) = 15x + 40<\/p>\n

Question 16.
\nClassify the equation 6(x + 2) = 5(x + 7) as having one solution, no solution, or infinitely many solutions.
\nAnswer:
\nThe given equation is:
\n6 (x + 2) = 5 (x + 7)
\nSo,
\n6 (x) + 6 (2) = 5 (x) + 5 (7)
\n6x + 12 = 5x + 35
\nRearrange the like terms
\nSo,
\n6x – 5x = 35 – 12
\nx = 23
\nHence, from the above,
\nWe can conclude that the given equation has only 1 solution<\/p>\n

Question 17.
\nSolve 6x + 14x + 5 = 5(4x + 1). Write a word problem that this equation, or any of its equivalent forms, represents.
\nAnswer:
\nThe given equation is:
\n6x + 14x + 5 = 5 (4x + 1)
\nSo,
\n20x + 5 = 5 (4x) + 5 (1)
\n20x + 5 = 20x + 5
\nSubtract with 20x on both sides
\nSo,
\n20x – 20x + 5 = 20x – 20x + 5
\n5 = 5
\nHence, from the above,
\nWe can conclude that the given equation is true for any value of x i..e, the given equation has infinitely many solutions<\/p>\n

Question 18.
\nClassify the equation 170x \u2013 1,000 = 30(5x \u2013 30) as having one solution, no solution, or infinitely many solutions.
\nAnswer:
\nThe given equation is:
\n170x – 1,000 = 30 (5x – 30)
\nSo,
\n170x – 1,000 = 30 (5x) – 30 (30)
\n170x – 1,000 = 150x – 900
\nRearrange the like terms
\nSo,
\n170x – 150x = 1,000 – 900
\n20x = 100
\nx= \\(\\frac{100}{20}\\)
\nx = 5
\nHence, from the above,
\nWe can conclude that the given equation has only 1 solution<\/p>\n

Question 19.
\nHigher Order Thinking<\/strong> Write one equation that has one solution, one equation that has no solution, and one equation that has infinitely many solutions.
\nAnswer:
\nThe example representation of the equation that has one solution is:
\n20x + 5 = 15x – 4
\nThe example representation of the equation that has no solutions is:
\n3 (6x – 2) = 9 (2x – 4)
\nThe example representation of the equation that has infinitely many solutions is:
\n4 (2x – 6) = 8 (x – 3)<\/p>\n

Question 20.
\nSolve 4(4x – 2) + 1 = 16x \u2013 7.
\nAnswer:
\nThe given equation is:
\n4 (4x – 2) + 1 = 16x – 7
\nSo,
\n4 (4x) – 4 (2) + 1 = 16x – 7
\n16x – 8 + 1 = 16x – 7
\n16x – 7 = 16x – 7
\nSubtract with 16x on both sides
\nSo,
\n16x – 16x – 7 = 16x – 16x – 7
\n-7 = -7
\n7 = 7
\nHence, from the above,
\nWe can conclude that the given equation is true for all the values of x i..e, the given equation has infinitely many solutions<\/p>\n

Question 21.
\nSolve 6x + 26x \u2013 10 = 8(4x + 10).
\nAnswer:
\nThe given equation is:
\n6x + 26x – 10 = 8 (4x + 10)
\nSo,
\n32x – 10 = 8 (4x) + 8 (10)
\n32x – 10 = 32x + 80
\nSubtract with 32x on both sides
\nSo,
\n32x – 32x – 10 = 32x – 32x + 80
\n-10 = 80
\nHence, from the above,
\nWe can conclude that the given equation has no solution<\/p>\n

Question 22.
\nClassify the equation 64x \u2013 16 = 16(4x \u2013 1) as having one solution, no solution, or infinitely many solutions.
\nAnswer:
\nThe given equation is:
\n64x – 16 = 16 (4x – 1)
\nSo,
\n64x – 16 = 16 (4x) – 16 (1)
\n64x – 16 = 64x – 16
\nSubtract with 64x on both sides
\nSo,
\n64x – 64x – 16 = 64x – 64x – 16
\n– 16 = -16
\n16 = 16
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

Question 23.
\nClassify the equation 5(2x + 3) = 3(3x + 12) as having one solution, no solution, or infinitely many solutions.
\nAnswer:
\nThe given equation is:
\n5 (2x + 3) = 3 (3x + 12)
\nSo,
\n5 (2x) + 5 (3) = 3 (3x) + 3 (12)
\n10x + 15 = 9x + 36
\nRearrange the like terms
\nSo,
\n10x – 9x = 36 – 15
\nx = 21
\nHence, from the above,
\nWe can conclude that the given equation has only 1 solution<\/p>\n

Assessment Practice<\/strong>
\nQuestion 24.
\nWhich of the following best describes the solution to the equation 4(2x + 3) = 16x + 12 – 8x?
\nA. The equation has one solution.
\nB. The equation has infinitely many solutions.
\nC. The equation has no solution.
\nD. The equation has two solutions.
\nAnswer:
\nThe given equation is:
\n4 (2x + 3) = 16x + 12 – 8x
\nSo,
\n4 (2x) + 4 (3) = 4x + 12
\n8x + 12 = 4x + 12
\nRearrange the like terms
\nSo,
\n8x – 4x = 12 – 12
\n4x = 0
\nx = 0
\nSo,
\nThe given equation has only 1 solution
\nHence, from the above,
\nWe can conclude that option A matches with the solution of the given equation<\/p>\n

Question 25.
\nWhich of the following statements are true about the equation 10x + 45x \u2013 13 = 11(5x + 6)? Select all that apply.
\n\u2610 The operations that can be used to solve the equation are addition and multiplication.
\n\u2610 The operations that can be used to solve the equation are multiplication and division.
\n\u2610 The equation has infinitely many solutions.
\n\u2610 The equation has a solution of x = 53.
\n\u2610 The equation has no solution.
\nAnswer:
\nLet the given options be named: A, B, C, D, and E
\nNow,
\nThe given equation is:
\n10x + 45x – 13 = 11 (5x + 6)
\nSo,
\n55x – 13 = 11 (5x) + 11 (6)
\n55x – 13 = 55x + 66
\nSubtract with 55x on both sides
\nSo,
\n55x – 55x – 13 = 55x – 55x + 66
\n-13 = 26
\nHence, from the above,
\nWe can conclude that option A and option E matches with the situation for the given equation<\/p>\n

Topic 2 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary<\/strong> How can you determine the number of solutions for an equation? Lesson 2-4
\nAnswer:
\nA one-variable equation has infinitely many solutions when solving results in a true statement, such as 2 = 2.
\nA one-variable equation has one solution when solving results in one value for the variable, such as x = 2.
\nA one-variable equation has no solution when solving results in an untrue statement, such as 2 = 3.<\/p>\n

Question 2.
\nSolve the equation –\\(\\frac{2}{3}\\)d – \\(\\frac{1}{4}\\)d = -22 for d. Lesson 2-1
\nAnswer:
\nThe given equation is:
\n–\\(\\frac{2}{3}\\)d – \\(\\frac{1}{4}\\)d = -22
\nSo,
\n\\(\\frac{8 + 3}{12}\\)d = 22
\n\\(\\frac{11}{12}\\)d = 22
\nMultiply with \\(\\frac{12}{11}\\) on both sides
\nSo,
\nd = 22 \u00d7 \\(\\frac{12}{11}\\)
\nd = \\(\\frac{22 \u00d7 12}{11}\\)
\nd = 24
\nHence, from the above,
\nWe can conclude that the value of d is: 24<\/p>\n

Question 3.
\nEdy 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
\nAnswer:
\nIt is given that
\nEdy 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.
\nNow,
\nLet the number of months be x
\nSo,
\nThe amount of money in the account of Edy = $450 + $40x
\nThe amount of money in the account of Juan = $975 – $45.45x – $19.55x
\nNow,
\nTo find after how many months they will have the same amount of money in their accounts,
\n$450 + $40x = $975 – $45.45x – $19.55x
\n$450 + $40x = $975 – $65x
\nRearrange the like terms
\nSo,
\n$65x + $40x = $975 – $450
\n$105x = $525
\nDivide by 105 into both sides
\nSo,
\nx = \\(\\frac{$25}{105}\\)
\nx = 5 months
\nHence, from the above,
\nWe can conclude that after 5 months, Edy and Jian will have the same amount of money in their accounts<\/p>\n

Question 4.
\nWhich equation has infinitely many solutions? Lesson 2-4
\nA. \\(\\frac{3}{4}\\)x + x – 5 = 10 + 2x
\nAnswer:
\nThe given equation is:
\n\\(\\frac{3}{4}\\)x + x – 5 = 10 + 2x
\n\\(\\frac{3 + 4}{4}\\)x – 5 = 10 + 2x
\n\\(\\frac{7}{4}\\)x – 5 = 10 + 2x
\nRearrange the like terms
\nSo,
\n\\(\\frac{7}{4}\\)x – 2x = 10 + 5
\n–\\(\\frac{1}{4}\\)x = 15
\nMultiply with -4 on both sides
\nSo,
\nx = -60
\nHence, from the above,
\nWe acn conclude that the given equation has only 1 solution<\/p>\n

B. 3x – 2.7 = 2x + 2.7 + x
\nAnswer:
\nThe given equation is:
\n3x – 2.7 = 2x + x + 2.7
\n3x – 2.7 = 3x + 2.7
\nSubtract with 3x on both sides
\nSo,
\n-2.7 = 2.7
\nHence, from the above,
\nWe can conclude that the given equation has no solutions<\/p>\n

C. 9x + 4.5 \u2013 2x = 2.3 +7x + 2.2
\nAnswer:
\nThe given equation is:
\n9x + 4.5 – 2x = 2.3 + 7x + 2.2
\n7x + 4.5 = 7x + 4.5
\nSubtract with 7x on both sides
\nSo,
\n4.5 = 4.5
\nHence, from the above,
\nWe can conclude that the given equation has infinitely many solutions<\/p>\n

D. \\(\\frac{1}{5}\\) x – 7 = \\(\\frac{3}{4}\\) + 2x – 25\\(\\frac{3}{4}\\)
\nAnswer:
\nThe given equation is:
\n\\(\\frac{1}{5}\\) x – 7 = \\(\\frac{3}{4}\\) + 2x – 25\\(\\frac{3}{4}\\)
\nWe know that,
\n25\\(\\frac{3}{4}\\) = \\(\\frac{103}{4}\\)
\nSo,
\n\\(\\frac{1}{5}\\) x – 7 = \\(\\frac{3}{4}\\) + 2x – \\(\\frac{103}{4}\\)
\nRearrange the like terms
\nSo,
\n\\(\\frac{1}{5}\\)x – 2x = 7 – \\(\\frac{103}{4}\\)
\n–\\(\\frac{9}{5}\\)x = –\\(\\frac{75}{4}\\)
\nMultiply with –\\(\\frac{5}{9}\\) on both sides
\nSo,
\nx = \\(\\frac{75 \u00d7 5}{4 \u00d7 9}\\)
\nx = \\(\\frac{125}{4}\\)
\nHence, from the above,
\nWe can conclude that the given equation has only 1 solution<\/p>\n

Question 5.
\nSolve the equation -4(x – 1) + 6x = 2(17 \u2013 x) for x. Lesson 2.3
\nAnswer:
\nThe given equation is:
\n-4 (x – 1) + 6x = 2 (17 – x)
\nSo,
\n-4 (x) + 4 (1) + 6x = 2 (17) – 2 (x)
\n-4x + 4 + 6x = 34 – 2x
\n2x + 4 = 34 – 2x
\nRearrange the like terms
\nSo,
\n2x + 2x = 34 – 4
\n4x = 30
\nDivide by 4 on both sides
\nSo,
\nx = \\(\\frac{30}{4}\\)
\nx = \\(\\frac{15}{2}\\)
\nHence, from the above,
\nWe can conclude that the value of x is: \\(\\frac{15}{2}\\)<\/p>\n

Question 6.
\nHakeem 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
\nAnswer:
\nIt is given that
\nHakeem subtracted 8 from a number, then multiplied the difference by \\(\\frac{4}{5}\\). The result was 20.
\nNow,
\nLet the number be x
\nSo,
\nAccording to Hakeem,
\nThe expression that represents the given situation is:
\n\\(\\frac{4}{5}\\) (x – 8) = 20
\nMultiply with \\(\\frac{5}{4}\\) on both sides
\nSo,
\nx – 8 = \\(\\frac{5 \u00d7 20}{4}\\)
\nx – 8 = 25
\nAdd with 8 on both sides
\nSo,
\nx = 25 + 8
\nx = 34
\nHence, from the baove,
\nWe can conclude that Hakeem’s number is: 34<\/p>\n

Topic 2 MID-TOPIC PERFORMANCE TASK<\/h3>\n

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.<\/p>\n

PART A<\/strong>
\nWrite and solve an equation to find when Wanda will catch up to Hector.
\nAnswer:
\nIt is given that
\nHector 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.
\nNow,
\nLet the time be x
\nWe know that,
\nSpeed = \\(\\frac{Distance} {Time}\\)
\nSo,
\nTime = \\(\\frac{Distance}{Speed}\\)
\nNow,
\nTime taken by Hector to complete a bicycle race = \\(\\frac{42 – 18}{12}\\)
\nx = \\(\\frac{24}{12}\\)
\nx = 2 hours
\nNow,
\nTime taken by Wanda to complete the bicycle race = \\(\\frac{The total distance of race}{The speed traveled by Wanda}\\)
\nx = \\(\\frac{42}{16}\\)
\nx = \\(\\frac{21}{8}\\) hours
\nx = 2.625 hours
\nNow,
\nThe 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
\n= 2.625 – 2
\n= 0.625
\n= 0.625 (60 minutes)
\n= 37.5 minutes
\nHence, from the above,
\nWe can conclude that Wanda will catch up to Hector after 37.5 minutes of Hector completing the race<\/p>\n

PART B<\/strong>
\nWill Wanda catch up to Hector before the race is complete? Explain.
\nAnswer:
\nFrom part (a),
\nThe time taken by Hector to complete the race is: 2 hours
\nThe time taken by Wanda to complete the race is: 2.625 hours
\nSo,
\nFrom the above times,
\nWe can observe that the race is completed at 2 hours
\nHence, from the above,
\nWe can conclude that Wanda can’t catch up to Hector before the race is complete<\/p>\n

PART C<\/strong>
\nAt what constant speed could Wanda travel to catch up with Hector at the finish line? Explain.
\nAnswer:
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nSo,
\nThe 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}\\)
\n= \\(\\frac{42}{2}\\)
\n= 21 miles per hour
\nHence, from the above,
\nWe can conclude that at 21 miles per hour speed, Wanda could catch up to Hector<\/p>\n

3-Act Mathematical Modeling: Powering Down<\/h3>\n

3-ACT MATH<\/strong>
\n\"Envision<\/p>\n

\u0410\u0421\u0422 1<\/strong><\/p>\n

Question 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:
\nAfter watching the video,
\nThe first question that comes to mind is:
\nwhat will be the battery percentage you should have to complete your work?<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\n\"Envision
\nAnswer:
\nThe main question you will answer is:
\nwhat will be the battery percentage you should have to complete your work?<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Predict an answer to this Main Question. Explain your prediction.
\nAnswer:
\nThe answer to the main question is: 100%
\nReason for the prediction:
\nWe don’t know how much work has left. So, it is better to have a battery percentage of 100%<\/p>\n

Question 4.
\nOn the number line below, write a time that is too early to be the answer. Write a time that is too late.
\n\"Envision
\nAnswer:
\nThe time that is too early to be the answer for the above problem is: 5 minutes
\nThe time that is too late to be the answer for the above problem is: Greater than the time that battery percentage is 100%<\/p>\n

Question 5.
\nPlot your prediction on the same number line.<\/p>\n

ACT 2<\/strong><\/p>\n

Question 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision
\nAnswer:
\nThe information in this situation that would be helpful to know is:
\nA) The time is taken for battery percentage to be full<\/p>\n

Question 7.
\nUse Appropriate Tools<\/strong> What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 8.
\nModel with Math<\/strong>
\nRepresent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it earlier or later than your prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Question 10.
\nWrite the answer you saw in the video.
\nAnswer:<\/p>\n

Question 11.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\n\"Envision
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere<\/strong> Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

Act 3<\/strong><\/p>\n

Reflect<\/strong><\/p>\n

Question 13.
\nModel with Math<\/strong>
\nExplain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\n\"Envision
\nAnswer:<\/p>\n

Question 14.
\nLook for Relationships<\/strong> What pattern did you notice in the situation? How did you use that pattern?
\nAnswer:<\/p>\n

SEQUEL<\/strong><\/p>\n

Question 15.
\nBe Precise<\/strong> 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%.
\nAnswer:<\/p>\n

Lesson 2.5 Compare Proportional Relationships<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Mei Li is going apple picking. She is choosing between two places. The cost of a crate of apples at each place is shown.
\nWhere should Mei Li go to pick her apples? Explain.
\n\"Envision
\nAnswer:
\nIt is given that Mei Li is going apple picking. She is choosing between two places.
\nSo,
\nIn Annie’s Apple Orchard,
\nThe cost of 20lb of apples is: $7.25
\nIn Franklin’s fruit Orchard,
\nThe cost of 12lb of apples is: $5
\nWe know that,
\nWhere the cost of 1lb of apples is low, Mei Li will go there to buy the apples
\nNow,
\nIn Annie’s Apple Orchard,
\nThe cost of 1lb of apples = \\(\\frac{$7.25}{20}\\)
\n= $0.3625
\nIn Franklin’s fruit Orchard,
\nThe cost of 1lb of apples = \\(\\frac{$5}{12}\\)
\n= $0.4166
\nSo,
\nThe cost of 1lb of apples in Annie’s Orchard < The cost f 1lb of apples in Franklin’s fruit Orchard
\nHence, from the above,
\nWe can conclude that Mei Li should go to pick apples from Annie’s Apple Orchard<\/p>\n

Construct Arguments<\/strong>
\nWhat information provided can be used to support your answer?
\nAnswer:
\nFrom the given figure,
\nThe information provided that can be used to support your answer is:
\nThe weight of the apples is inversely proportional to the price of the apples
\nSo,
\nIn Annie’s Apple Orchard, the weight of the apples is high when compared to the weight of the apples in franklin’s fruit Orchard
\nHence,
\nThe price of the apples is low in Annie’s Apple Orchard when compared to Franklin’s fruit Orchard<\/p>\n

Focus on math practices<\/strong>
\nModel with Math<\/strong> Which representation did you use to compare prices? Explain why.
\nAnswer:
\nThe relation that is used to compare the prices of apples is:
\nWeight of the apples \u221d \\(\\frac{1}{Price of the apples}\\)<\/p>\n

? Essential Question<\/strong>
\nHow can you compare proportional relationships represented in different ways?
\nAnswer:
\nTo compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.<\/p>\n

Try It!<\/strong>
\nThe 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe 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.
\nSo,
\nThe rate that birds made by Josh = \\(\\frac{y}{x}\\)
\n= 2.5
\nNow,
\nFrom the graph,
\nThe rate that birds made by Marlo = \\(\\frac{Time taken to make birds by Marlo}{The number of birds}\\)
\n= \\(\\frac{40}{8}\\) (Here, we can take any value that is present in the graph. For example,\\(\\frac{20}{4}\\), \\(\\frac{10}{2}\\) etc., )
\n= 5
\nSo,
\nThe rate that birds made by Marlo > The rate that birds made by Josh
\nHence, from the above,
\nWe can conclude that Marlo makes birds at a faster rate<\/p>\n

Convince Me!<\/strong>
\nIf you were to graph the data for Josh and Marlo on the same coordinate plane, how would the two lines compare?
\nAnswer:
\nWhen we graph the data for Josh and Marlo on the same coordinate plane,
\nWe can observe that the two graphs will be the lines that are parallel to each other and the rate of change\u00a0 of Marlo will be greater than the rate of change of Josh<\/p>\n

Try It!<\/strong>
\nThe 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe 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.
\nNow,
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nSo,
\nThe speed of the fastest high-speed train in the United states = 600 \/ \\(\\frac{5}{2}\\)
\nWe know that,
\n2\\(\\frac{1}{2}\\) = \\(\\frac{5}{2}\\)
\nSo,
\nThe speed of the fastest high-speed train in the United states = \\(\\frac{600 \u00d7 2}{5}\\)
\n= 240 kilometers per hour
\nNow,
\nFrom the given graph,
\nThe speed of the fastest high-speed train in Japan = \\(\\frac{The difference between any two distances from the graph}{The\u00a0 difference between the values of the time that corresponds to the taken value of distances}\\)
\n= \\(\\frac{1000 – 650}{3 – 2}\\)
\n= 350 kilometers per hour
\nSo,
\nThe speed of the fastest high-speed train in Japan > The speed of the fastest high-speed train in the United States
\nHence, from the above,
\nWe can conclude that the fastest high-speed train in Japan has a maximum speed<\/p>\n

KEY CONCEPT<\/strong>
\nTo compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.
\nThe representations below show the rental cost per hour for canoes at three different shops.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How can you compare proportional relationships represented in different ways?
\nAnswer:
\nTo compare proportional relationships represented in different ways, find the unit rate, or the constant of proportionality, for each representation.<\/p>\n

Question 2.
\nHow can you find the unit rate or constant of proportionality for a relationship represented in a graph?
\nAnswer:
\nIn a graph,
\nThe unit rate or constant of proportionality for a relationship is represented by:
\n\\(\\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}\\)<\/p>\n

Question 3.
\nGeneralize<\/strong> Why can you use the constant of proportionality with any representation?
\nAnswer:
\nWe can use the constant of proportionality to find the rate of change between the physical quantities that have a proportional relationship
\nEx:
\nSpeed Vs Distance, Speed Vs Time, etc<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nAmanda 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.
\n\"Envision
\nWho earns more per hour?
\nAnswer:
\nIt is given that
\nAmanda 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.
\nNow,
\nThe Earnings per hour of Amanda = \\(\\frac{24 – 0}{4 – 0}\\)
\n= \\(\\frac{24}{4}\\)
\n= 6
\nThe Earnings per hour of Petra = \\(\\frac{15}{3}\\)
\n= 5
\nSo,
\nThe Earnings per hour of Amanda > The Earnings per hour of Petra
\nHence, from the above,
\nWe can conclude that Amanda earns more<\/p>\n

Question 5.
\nMilo 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nMilo 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.
\nSo,
\nNow,
\nThe 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}\\)
\n= \\(\\frac{5}{1}\\)
\n= $5
\nSo,
\nThe cost per pound of food at Pat’s Pet Palace < The cost per pound of food at Mark’s Mutt Market
\nHence, from the above,
\nWe can conclude that at Pat’s Pet Palace, Milo will pay a lower price per pound for dog food<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> For 6 and 7, complete the information to compare the rates.<\/p>\n

Question 6.
\nSam 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.
\n\"Envision
\n\"Envision
\nFind the unit rate (constant of proportionality) for Bobby.
\nUse (\"Envision) and (\"Envision) to find the constant of proportionality.
\nThe unit rate (constant of proportionality) is \"Envision
\nSo \"Envision cycled faster.
\nAnswer:
\nIt is given that
\nSam 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.
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nSo,
\nFor Sam, from the table,
\nSpeed = \\(\\frac{20}{2}\\) miles per hour
\n= 10 miles per hour
\nNow,
\nFor Bobby, from the graph,
\nSpeed = \\(\\frac{Any value of the distance from the graph}{The value of time that corresponds to the distance that we have taken}\\)
\n= \\(\\frac{72}{8}\\)
\n= 9 miles per hour
\nSo,
\nThe speed of Sam > The speed of Bobby
\nHence, from the above,
\nWe can conclude that Sam cycled faster<\/p>\n

Question 7.
\nModel with Math<\/strong> 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?
\n\"Envision
\nPauli’s Pizzeria takes in \"Envision per pizza.
\nLeo’s Pizzeria takes in \"Envision per pizza.
\n\"Envision‘s Pizzeria takes in more money per pizza.
\nAnswer:
\nIt is given that
\nThe 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
\nSo,
\nThe money earned by Pauli’s Pizzeria = \\(\\frac{y}{x}\\)
\n= 15 (From the given equation y = 15x)
\nNow,
\nFrom the given graph,
\nThe 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}\\)
\n= \\(\\frac{96}{8}\\)
\n= 12
\nSo,
\nThe money earned by Pauli’s Pizzeria > The money earned by Leo’s Pizzeria
\nHence, from the above,
\nWe can conclude that Pauli’s Pizzeria takes in more money per pizza<\/p>\n

Question 8.
\nThe 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the amount of savings over time in Eliana’s account. Lana, meanwhile, puts $50 each week into her savings account
\nSo,
\nThe 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}\\)
\n= \\(\\frac{94}{2}\\)
\n= $47
\nSo,
\nThe amount of savings over time in Elina’s account > The amount of savings over time in Lana’s account
\nHence, from the above,
\nWe can conclude that Elina is saving money at a greater rate<\/p>\n

Question 9.
\nMake Sense and Persevere<\/strong> 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.
\na. In order to compare the proportional relationships, what quantities should you use to find the unit rate?
\nAnswer:
\nIn order to compare the proportional relationships,
\nThe quantities you should use to find the unit rate is:
\nA) The number of miles walked
\nB) The amount of money raised for the corresponding number of miles<\/p>\n

b. Compare the amount of money raised per mile by the three people.
\n\"Envision
\nAnswer:
\nIt is given that
\nBeth, 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.
\nSo,
\nThe amount of money raised by Beth = \\(\\frac{y}{x}\\)
\n= $20 (From the equation y = 20x)
\nNow,
\nThe 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}\\)
\n= \\(\\frac{$45}{3}\\)
\n= $15
\nSo,
\nThe amount of money raised by Beth > The amount of money raised by Manuel = The amount of money raised by Petra
\nHence, from the above,
\nWe can conclude that Beth raised more amount of money when compared to Manuel and Petra<\/p>\n

Question 10.
\nHigher-Order Thinking<\/strong> 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.
\n\"Envision
\na. Do you agree with Winston? Explain your response.
\nAnswer:
\nIt is given that
\nWinston 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.
\nSo,
\nFrom the given information,
\nThe height growth of plant 1 > The height growth of plant 2
\nHence, from the above,
\nYou can agree with Winston<\/p>\n

b. What errors might Winston have made?
\nAnswer:
\nFor plant 1,
\nThe height growth per day = \\(\\frac{Any value of height}{The value of days correspond to the value of height}\\)
\n= \\(\\frac{6}{2}\\)
\n= 3 cm
\nFor plant 2,
\nThe height growth per day = \\(\\frac{Any value of height}{The value of days correspond to the value of height}\\)
\n= \\(\\frac{4}{2}\\)
\n= 2 cm
\nBut,
\nIt is given that
\nWinston says that Plant 1 grows 6 cm per day and Plant 2 grows 4 cm per day
\nBut according to the calculation,
\nPlant 1 grows 3 cm per day and plant 2 grows 2 cm per day
\nSo,
\nThe calculation of the height growth of the plants are the errors made by Winston<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 11.
\nAshton, 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nAshton, 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.
\nSo,
\nThe rate at which Ashton can type = \\(\\frac{y}{x}\\)
\n= 39 words per minute (From the equation y = 39x)
\nThe 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}\\)
\n= \\(\\frac{78}{2}\\)
\n= 39 words per minute
\nThe 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}\\)
\n= \\(\\frac{78}{2}\\)
\n= 39 words per minute
\nSo,
\nThe rate at which Ashton can type = The rate at which Alexa can type = the rate at which Clara can type
\nHence, from the above,
\nWe can conclude that no one is the fastest<\/p>\n

Lesson 2.6 Connect Proportional Relationships and Slope<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Solve & Discuss It!<\/strong><\/p>\n

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?
\n\"Envision
\nAnswer:
\nIt is given that
\nIn 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.
\nSo,
\nThe amount of money paid for 1 game = \\(\\frac{The amount of money paid for 5 games}{5}\\)
\n= \\(\\frac{$98.50}{5}\\)
\n= $19.70
\nSo,
\nThe amount of money paid for 14 games to Rashida = (The total number of games) \u00d7 (The amount of money paid for 1 game)
\n= 14 \u00d7 $19.70
\n= $275.80
\nHence, from the above,
\nWe 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<\/p>\n

Look for Relationships<\/strong>
\nHow is the number of games Rashida works related to her earnings?
\nAnswer:
\nFrom the above,
\nWe can observe that Rashida earns more money by refereeing more soccer games
\nHence, from the above,
\nWe can conclude that
\nThe number of games Rashida works \u221d The earnings of Rashida<\/p>\n

Focus on math practices<\/strong>
\nReasoning: How would Rashida’s earnings change if she were paid by the hour instead of by the game?
\nAnswer:
\nRashida’s earnings would increase if she were paid by the hour instead of by the game
\nExample:
\nFrom the above,
\nWe can observe that
\nThe money earned by Rashida per game = $17.90
\nBut, 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
\nThe amount of money earned by Rashida for this game = $17.90 \u00d7 2 = $35.80
\nHence, from the above,
\nWe can conclude that Rashida can earn more if she were paid by the hour instead of by the game<\/p>\n

? Essential Question<\/strong>
\nWhat is the slope?
\nAnswer:
\nThe 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).
\nThe representation of the slope mathematically is:
\nSlope = \\(\\frac{Rise}{Run}\\)<\/p>\n

Try It!<\/strong><\/p>\n

Jack graphs how far he plans to bike over a 3-day charity ride. Find the slope of the line.
\n\"Envision
\n\"Envision
\nAnswer:
\nIt is given that Jack graphs how far he plans to bike over a 3-day charity ride
\nNow,
\nFrom the given graph,
\nThe given points are: (3, 90), and (2, 60)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1}{x2 – x1}\\)
\nSo,
\nThe slope of the line = \\(\\frac{60 – 90}{2 – 3}\\)
\n= 30
\nHence, from the above,
\nWe can conclude that the slope of the line is: 30<\/p>\n

Convince Me!<\/strong>
\nHow do the unit rate and constant of proportionality relate to the slope of a line?
\nAnswer:
\nThe 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.<\/p>\n

Try It!<\/strong>
\nThe 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the proportions of red and blue food coloring that Taylor mixes to make the purple frosting.
\nNow,
\nFrom the given graph,
\nThe given points are: (50, 70), and (25, 35)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1}{x2 – x1}\\)
\nSo,
\nThe slope of the line = \\(\\frac{70 – 35}{50 – 25}\\)
\n= \\(\\frac{35}{25}\\)
\n= \\(\\frac{7}{5}\\)
\nHence, from the above,
\nWe can conclude that
\nFor every 7 parts of red food coloring, we have to mix 5 parts of blue food coloring to make the purple frosting<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

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.
\n\"Envision
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nWhat is the slope?
\nAnswer:
\nThe 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).
\nThe representation of the slope mathematically is:
\nSlope = \\(\\frac{y2\u00a0<\/sub> – y1}{x2 – x1}\\) (or) Sloe = \\(\\frac{Rise}{Run}\\)<\/p>\n

Question 2.
\nReasoning<\/strong> How is the slope related to a unit rate?
\nAnswer:
\nThe 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.<\/p>\n

Question 3.
\nLook for Relationships<\/strong> Why is the slope between any two points on a straight line always the same?
\nAnswer:
\nThe ratio\u00a0<\/span>of the\u00a0rise over run describes the\u00a0<\/span>slope of\u00a0all\u00a0<\/span>straight lines. This ratio is constant\u00a0<\/span>between any two points\u00a0along a\u00a0<\/span>straight line, which means\u00a0<\/span>that\u00a0the\u00a0<\/span>slope of\u00a0a\u00a0<\/span>straight line\u00a0is constant, too,\u00a0<\/span>no\u00a0matter where it is measured along the\u00a0<\/span>line.<\/span><\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nWhat is the slope of the line?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\\)
\n= \\(\\frac{Price ($)}{Grapes (lb)}\\)
\nSo,
\nThe slope of the line = \\(\\frac{6}{2}\\)
\n= 3
\nHence, from the above,
\nWe can conclude that the slope of the line is: 3<\/p>\n

Question 5.
\nThe scale of a model airplane is shown in the graph.
\na. Find the slope of the line using \\(\\frac{y2\u00a0<\/sub> – y1}{x2 – x1}\\)
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the given graph,
\nThe given points are: (6, 10), and (3, 5)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1}{x2 – x1}\\)
\nSo,
\nThe slope of the line = \\(\\frac{5 – 10}{3 – 6}\\)
\n= \\(\\frac{5}{3}\\)
\nHence, from the above,
\nWe can conclude that the slope of the line is: \\(\\frac{5}{3}\\)<\/p>\n

b. What does the slope mean in the problem situation?
\nAnswer:
\nFrom part (a),
\nThe slope is: \\(\\frac{5}{3}\\)
\nSo,
\nFrom the above slope,
\nWe can conclude that for every 3 cm, the model airplane can fly 5 feet<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice in 6 and 7, find the slope of each line.<\/p>\n

Question 6.
\nThe 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.
\n\"Envision
\n\"Envision
\nThe machine \"Envision can make soda bottles each minute.
\nAnswer:
\nIt is given that
\nThe graph shows the number of soda bottles a machine can make over time
\nNow,
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the given graph,
\nThe given points are: (6, 150), and (2, 50)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1}{x2 – x1}\\)
\nSo,
\nThe slope of the line = \\(\\frac{50 – 150}{2 – 6}\\)
\n= \\(\\frac{100}{4}\\)
\n= 25
\nHence, from the above,
\nWe can conclude that the machine can make 25 soda bottles each minute<\/p>\n

Question 7.
\n<\/span>Find the slope of the line.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\\)
\n= \\(\\frac{Items}{Time in min}\\)
\nSo,
\nThe slope of the line = \\(\\frac{50}{10}\\)
\n= 5
\nHence, from the above,
\nWe can conclude that the slope of the line is: 5<\/p>\n

Question 8.
\nReasoning<\/strong> How can you find the slope of the line that passes through the points (0,0) and (2, 4)? Explain.
\nAnswer:
\nThe given points are: (0, 0), and (2, 4)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\nSo,
\nThe slope of the line = \\(\\frac{4 – 0}{2 – 0}\\)
\n= \\(\\frac{4}{2}\\)
\n= 2
\nHence, from the above,
\nWe can conclude that the slope of the line is: 2<\/p>\n

Question 9.
\nThe 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?
\nAnswer:
\nIt is given that the points (2.1, -4.2) and (2.5, -5) form a proportional relationship
\nNow,
\nThe given points are: (2.1, -4.2), and (2.5, -5)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\nSo,
\nThe slope of the line = \\(\\frac{-5 + 4.2}{2.5 – 2.1}\\)
\n= \\(\\frac{-0.8}{0.4}\\)
\n= -2
\nHence, from the above,
\nWe can conclude that the slope of the line that passes through the given points is: -2<\/p>\n

Question 10.
\nFind the slope of the line.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the graph,
\nWe can observe that
\nThe given points are: (-3, 7), and (-1, 2)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\nSo,
\nThe slope of the line = \\(\\frac{2 – 7}{-1 + 3}\\)
\n= \\(\\frac{-5}{2}\\)
\n= –\\(\\frac{5}{2}\\)
\nHence, from the above,
\nWe can conclude that the slope of the line that passes through the given points is: –\\(\\frac{5}{2}\\)<\/p>\n

Question 11.
\nThe graph shows the number of Calories Natalia burned while running.
\na. What is the slope of the line?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\\)
\n= \\(\\frac{Calories}{Time in min}\\)
\nSo,
\nThe slope of the line = \\(\\frac{70}{7}\\)
\n= 10
\nHence, from the above,
\nWe can conclude that the slope of the line is: 10<\/p>\n

b. What does the slope tell you?
\nAnswer:
\nFrom part (a),
\nWe can observe that
\nThe slope of the line is: 10
\nSo,
\nFrom the given slope,
\nWe can conclude that Natalia burns 10 calories per minute while running<\/p>\n

Question 12.
\nCritique Reasoning<\/strong> 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.
\na. What is the speed of the car?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{ Any value of y}{The value of x that corresponds to the taken value of y}\\)
\n= \\(\\frac{Distance in miles}{Time in hours}\\)
\nSo,
\nThe slope of the line = \\(\\frac{256}{4}\\)
\n= 64
\nHence, from the above,
\nWe can conclude that the speed of the car is: 64 miles per hour<\/p>\n

b. What error might Anna have made?
\nAnswer:
\nFrom part (a),
\nWe can observe that the speed of the car is: 64 miles per hour
\nBt,
\nAccording to Anna,
\nThe speed of the car is: \\(\\frac{1}{64}\\) miles per hour
\nSo,
\nThe 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}\\)<\/p>\n

Question 13.
\nHigher-Order Thinking<\/strong> 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nYou 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)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\)
\nSo,
\nFrom the given information,
\nWe can write the slope as:
\nSlope = \\(\\frac{11}{5}\\)
\nNow,
\nCompare the given point with (x, y)
\nSo,
\nThe slope of the line = \\(\\frac{y}{x}\\)
\n= \\(\\frac{y}{10}\\)
\nSo,
\n\\(\\frac{y}{10}\\) = \\(\\frac{11}{5}\\)
\nMultiply with 10 on both sides
\nSo,
\ny = \\(\\frac{11 \u00d7 10}{5}\\)
\ny = 22
\nHence, from the above,
\nWe can conclude that the value of y is: 22<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 14.
\nThe points (15, 21) and (25, 35) form a proportional relationship.
\na. Find the slope of the line that passes through these points.
\nAnswer:
\nIt is given that the points (15, 21) and (25, 35) form a proportional relationship.
\nNow,
\nThe given points are: (15, 21), and (25, 35)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\) = \\(\\frac{y2\u00a0<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\nSo,
\nThe slope of the line = \\(\\frac{35 – 21}{25 – 15}\\)
\n= \\(\\frac{14}{10}\\)
\n= \\(\\frac{7}{5}\\)
\nHence, from the above,
\nWe can conclude that the slope of the line that passes through the given points is: \\(\\frac{7}{5}\\)<\/p>\n

b. Which graph represents this relationship?
\n\"Envision
\nAnswer:
\nWe know that,
\nThe representation of the equation when two points form\u00a0 a proportionate relationship is:
\ny = kx
\nand the line have to pass through the origin i.e., (0, 0)
\nSo,
\nFrom the given graphs,
\nThe graphs B and C have the possibility to become the graph of the given points
\nNow,
\nWe know that,
\nSlope = \\(\\frac{Rise}{Run}\\)
\nFrom graph B,
\nSlope = \\(\\frac{42}{30}\\)
\n= \\(\\frac{7}{5}\\)
\nFrom graph C,
\nSlope = \\(\\frac{30}{42}\\)
\n= \\(\\frac{5}{7}\\)
\nHence, from the above,
\nWe can conclude that the graph B represents the given relationship<\/p>\n

Lesson 2.7 Analyze Linear Equations: y = mx<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explore It!<\/strong><\/p>\n

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.
\n\"Envision
\nA. What representation can show the distance the car will travel over time?
\nAnswer:
\nIt is given that
\nA 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.
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nIt is given that speed is constant
\nSo,
\nDistance \u221d Time
\nSo,
\nThe greater the distance, the greater the time
\nHence, from the above,
\nWe can conclude that
\nThe representation that can show the distance the car will travel over time is:
\nDistance \u221d Time<\/p>\n

B. What expression can show the distance the car will travel over time?
\nAnswer:
\nFrom part (a),
\nWe can observe that
\nDistance \u221d Time (Since the speed is constant)
\nHence,
\nThe expression that can show the distance the car will travel over time is:
\nDistance = k (Time)
\nWhere,
\nk is a constant<\/p>\n

C. Compare the representation and the expression. Which shows the distance traveled over time more clearly? Explain.
\nAnswer:
\nFrom part (a),
\nThe representation that can show the distance traveled over time is:
\nDistance \u221d Time
\nThe expression that can show the distance traveled over time is:
\nDistance = k (Time)
\nNow,
\nFrom the representation and the expression,
\nWe 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
\nHence, from the above,
\nWe can conclude that the expression shows the distance traveled over time more clearly<\/p>\n

Focus on math practices<\/strong>
\nBe Precise How would the representation or expression change if the speed was converted to miles per minute?
\nAnswer:
\nFrom part (a),
\nThe representation is:
\nDistance \u221d Time
\nThe expression is:
\nDistance = k (Time)
\nNow,
\nEven 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<\/p>\n

? Essential Question<\/strong>
\nHow does slope relate to the equation for a proportional relationship?
\nAnswer:
\nThe steepness of the\u00a0slope\u00a0for directly\u00a0proportional relationships\u00a0increases as the value of the constant m (y = mx) increases.<\/p>\n

Try It!<\/strong>
\nWrite an equation to describe the relationship shown in the graph.
\n\"Envision. The equation of the line is y = \"Envisionx.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the given graph,
\nThe 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]
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope can be represented as “m”
\nSo,
\nm = \\(\\frac{y2\u00a0<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\nSo,
\nm = \\(\\frac{80 – 60}{4 – 3}\\)
\n= \\(\\frac{20}{1}\\)
\n= 20
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nHence, from the above,
\nWe can conclude that
\nThe equation of the line is: y = 20x<\/p>\n

Convince Me!<\/strong>
\nHow do the equations y = mx and y = kx compare?
\nAnswer:
\nWe 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).<\/p>\n

Try It!<\/strong>
\na. Write the equation of the line.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the given graph,
\nThe points are: (10, 4), and (-10, -4)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope can be represented as “m”
\nSo,
\nm = \\(\\frac{y2\u00a0<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\nSo,
\nm = \\(\\frac{-4 – 4}{-10 – 10}\\)
\n= \\(\\frac{-8}{-20}\\)
\n= \\(\\frac{2}{5}\\)
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nSo,
\ny = \\(\\frac{2}{5}\\)x
\nMultiply with 5 on both sides
\nSo,
\n5y = 2x
\nHence, from the above,
\nWe can conclude that
\nThe equation of the line is: 5y = 2x<\/p>\n

b. Graph the line y = -3x.
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = -3x
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

The equation for a proportional relationship is y = mx where m represents the slope of the line.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> How does slope relate to the equation for a proportional relationship?
\nAnswer:
\nThe steepness of the\u00a0slope\u00a0for directly\u00a0proportional relationships\u00a0increases as the value of the constant m (y = mx) increases.<\/p>\n

Question 2.
\nLook for Relationships<\/strong> What do the graphs of lines in the form y = mx have in common? How might they differ?
\nAnswer:
\nThe graphs of lines in the form y = mx are all straight lines that pass through the origin<\/p>\n

Question 3.
\nUse Structure<\/strong> The table below shows the distance a train traveled over time. How can you determine the equation that represents this relationship?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe table below shows the distance a train traveled over time.
\nNow,
\nVerify whether \\(\\frac{Distance}{Time}\\) is constant or not
\nNow,
\nFrom the given table,
\nFor 25 m and 2s,
\n\\(\\frac{Distance}{Time}\\) = \\(\\frac{25}{2}\\)
\nFor 50m and 4s,
\n\\(\\frac{Distance}{Time}\\) = \\(\\frac{50}{4}\\) = \\(\\frac{25}{2}\\)
\nSince,
\n\\(\\frac{Distance}{Time}\\) is constant
\nSpeed is also constant
\nSo,
\nThe representation of the equation that describes the given relationship is:
\nDistance = k (Time)
\nSo,
\ny = mx [ Compare the above equation with y = mx ]
\nWhere
\nm is a constant slope
\nSo,
\ny = \\(\\frac{25}{2}\\)x
\n2y = 25x
\nHence, from the above,
\nWe can conclude that the representation of the equation that represents the given situation is: 2y = 25x<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nThe relationship between a hiker’s elevation and time is shown in the graph.
\n\"Envision
\na. Find the constant of proportionality of the line. Then find the slope of the line.
\nAnswer:
\nIt is given that
\nThe relationship between a hiker’s elevation and time is shown in the graph.
\nNow,
\nWe know that,
\nThe constant of proportionality and the slope are the same
\nSo,
\nSlope of the line (m) = \\(\\frac{y}{x}\\)
\nSo,
\nFrom the given graph,
\n\\(\\frac{y}{x}\\) = \\(\\frac{120}{4}\\)
\n= 30
\nSo,
\nm = 30
\nHence, from the above,
\nWe can conclude that the slope of the line is: 30<\/p>\n

b. Write the equation of the line.
\nAnswer:
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nFrom part (a),
\nm = 30
\nHence, from the above,
\nWe can conclude that the equation of the line is: y = 30x<\/p>\n

Question 5.
\nGraph the equation y = –\\(\\frac{1}{2}\\)x.
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = –\\(\\frac{1}{2}\\)x
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nLeveled Practice<\/strong> 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.
\na. Use two sets of coordinates to write an equation to describe the relationship.
\n\"Envision
\n\"Envision
\nAnswer:
\nIt is given that
\nResting 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.
\nNow,
\nFrom the given graph,
\nThe points are: (3, 210), and (4, 280)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), and (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope can be represented as “m”
\nSo,
\nm = \\(\\frac{y2\u00a0<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\nSo,
\nm = \\(\\frac{280 – 210}{4 – 3}\\)
\n= \\(\\frac{70}{1}\\)
\n= 70
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nHence, from the above,
\nWe can conclude that
\nThe equation of the line that describes the given situation is: y = 70x<\/p>\n

b. Interpret the equation in words.
\nThe heart’s resting heart rate is \"Envision beats each minute.
\nAnswer:
\nFrom part (a),
\nThe equation of the line that describes the given situation is: y = 70x
\nHence, from the above,
\nWe can conclude that the heart’s resting heart rate is 70 beats each minute<\/p>\n

Question 7.
\nModel with Math<\/strong> 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph relates the number of gallons of white paint to the number of gallons of red paint Jess used to make the perfect pink.
\nNow,
\nThe given graph is:
\n\"Envision
\nNow,
\nSlope of the given line (m) = \\(\\frac{y}{x}\\)
\nm = \\(\\frac{4}{1}\\)
\nm = 4
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nHence, from the above,
\nWe can conclude that the equation of the line that represents the given situation is: y = 4x<\/p>\n

Question 8.
\nCritique Reasoning<\/strong> Franco made this graph to show the equation y = -x. Is the graph correct? Explain.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nWe know that,
\nSlope of the line (m) = \\(\\frac{y}{x}\\)
\nm = \\(\\frac{4}{4}\\)
\nm = 1
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nSo,
\nThe equation of the line is:
\ny = x
\nBut,
\nFranco made this graph to show the equation y = -x
\nHence, from the above,
\nWe can conclude that the graph of Franco is not correct<\/p>\n

Question 9.
\nThe graph shows a proportional relationship between the variables x and y.
\na. Write an equation to model the relationship.
\nb. Reasoning<\/strong> Explain how you know if an equation or a graph represents a proportional relationship.
\n\"Envision
\nAnswer:
\na.
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe slope of the line (m) = \\(\\frac{y}{x}\\)
\n= \\(\\frac{96}{8}\\)
\n= 12
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nSo,
\nThe equation of the line to the given relationship is:
\ny = 12x
\nHence, from the above,
\nWe can conclude that the equation of the line that represents the given situation is: y = 12x<\/p>\n

b.
\nIf the relationship\u00a0between two quantities is a\u00a0proportional relationship, this\u00a0relationship\u00a0can be represented by\u00a0the graph\u00a0of a straight line through\u00a0the\u00a0origin with a slope equal\u00a0to the\u00a0unit rate. For each point (x, y) on\u00a0the graph, \u017e is equal\u00a0to\u00a0k, where k is\u00a0the\u00a0unit rate.<\/p>\n

Question 10.
\nModel with Math<\/strong> Graph the equation y = -5x on the coordinate plane.
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = -5x
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

Question 11.
\nGraph the equation y = \\(\\frac{3}{5}\\)x on the coordinate plane.
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = \\(\\frac{3}{5}\\)x
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

Question 12.
\nHigher-Order Thinking<\/strong> A movie theater sends out a coupon for 70% off the price of a ticket.
\na. Write an equation for the situation, where y is the price of the ticket with the coupon and x is the original price.
\n\"Envision
\nAnswer:
\nIt is given that
\nA movie theater sends out a coupon for 70% off the price of a ticket.
\nSo,
\nThe proportionality constant of the given situation = \\(\\frac{70}{100}\\)
\n= \\(\\frac{7}{10}\\)
\nWe know that,
\nProportionality constant = Slope
\nSo,
\nSlope (m) = \\(\\frac{7}{10}\\)
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nSo,
\ny = \\(\\frac{7}{10}\\)
\n10y = 7x
\nHence, from the above,
\nWe can conclude that the equation of the line for the given situation is: 10y = 7x<\/p>\n

b. Graph the equation and explain why the line should only be in the first quadrant.
\nAnswer:
\nFrom part (a),
\nThe equation of the line is:
\n10y = 7x
\nSo,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"
\nFrom the graph,
\nWe can observe that
\nThe graph should only be in 1st quadrant because the values of x and y are both positive<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 13.
\nAn equation and a graph of proportional relationships are shown. Which has the greater unit rate? y = \\(\\frac{47}{2}\\)x
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nSlope (m) = \\(\\frac{y}{x}\\)
\n= \\(\\frac{282}{6}\\)
\n= 47
\nNow,
\nThe given equation is:
\ny = \\(\\frac{47}{2}\\)x
\nSo,
\nSlope (m) = \\(\\frac{y}{x}\\)
\n= \\(\\frac{47}{2}\\)
\nNow,
\nWhen we compare the rates or slopes,
\n47 > \\(\\frac{47}{2}\\)
\nHence, from the above,
\nWe ca conclude that the unit rate of the graph is greater than the unit rate of the equation<\/p>\n

Question 14.
\nCar X travels 186 miles in 3 hours.
\nPART A<\/strong> Write the equation of the line that describes the relationship between distance and time.
\nAnswer:
\nIt is given that car X travels 186 miles in 3 hours.
\nNow,
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nWe know that,
\nThe equation of the line is:
\ny = mx
\nWhere,
\nm = \\(\\frac{Distance}{Time}\\)
\nSo,
\nThe equation of the line is:
\ny = \\(\\frac{186}{3}\\)x
\ny = 62x
\nHence, from the above,
\nWe can conclude that the equation of the line that descries the relationship between distance and time is:
\ny = 62x<\/p>\n

PART B<\/strong> Which graph represents the relationship between distance and time for Car X?
\n\"Envision
\nAnswer:
\nFrom part (a),
\nThe equation of the line that describes the relationship between distance and time is:
\ny = 62x
\nWhere,
\n62 —-> The value of \\(\\frac{y}{x}\\) (or) m
\nSo,
\nFrom the above graphs,
\nWe can observe that,
\nm = 62 is possible from graphs C and D
\nBut,
\nWe know that,
\nThe equation y = mx passes through the origin
\nHence, from the above,
\nWe can conclude that the graph C represents the relationship between distance and time for car X<\/p>\n

Lesson 2.8 Understand the y-Intercept of a Line<\/h3>\n

Solve and Discuss It!<\/strong><\/p>\n

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?
\n\"Envision
\nAnswer:
\nIt is given that
\nEight-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.
\nSo,
\nWhen Alex is 8 years old,
\nThe age of the horse in human years is: 50 years
\nNow,
\nFor every 2 human years, the horse ages 5 years
\nSo,
\nSo,
\nFor Alex,
\nThe number of times his age increases = \\(\\frac{8}{2}\\)
\n= 4 times
\nSo,
\nThe increase in the age of the horse when Alex is 8 years old = 5 \u00d7 4 = 20 years
\nSo,
\nThe age of the horse when Alex born = The present age of the horse – The increased age of the horse
\n= 50 – 20
\n= 30 years
\nHence, from the above,
\nWe can conclude that the age of the horse when Alex is born is: 30 years<\/p>\n

Focus on math practices<\/strong>
\nUse Structure<\/strong> 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?
\nAnswer:
\nIt is given that
\nA veterinarian says that a cat ages 8 years for every 2 human years.
\nNow,
\nLet the age of the cat in human years be x
\nSo,
\n\\(\\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}\\)
\n\\(\\frac{64}{x}\\) = \\(\\frac{8}{2}\\)
\nDivide by 64 into both sides
\nSo,
\n\\(\\frac{64}{x \u00d7 64}\\) = \\(\\frac{8}{2 \u00d7 64}\\)
\n\\(\\frac{1}{x}\\) = \\(\\frac{1}{16}\\)
\nx = 16 years
\nHence, from the above,
\nWe can conclude that the age of cat in human years is: 16 years<\/p>\n

? Essential Question<\/strong>
\nWhat is the y-intercept and what does it indicate?
\nAnswer:
\nThe\u00a0slope\u00a0and\u00a0y-intercept\u00a0values\u00a0indicate\u00a0characteristics of the relationship between the two variables x and\u00a0y. The\u00a0slope indicates\u00a0the rate of change in\u00a0y\u00a0per unit change in x. The\u00a0y-intercept indicates the y-value when the x-value is 0.<\/p>\n

Try It!<\/strong>
\nPrices 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 \"Envision
\nThe y-intercept is \"Envision
\n\"Envision
\nAnswer:
\nIt is given that
\nThe prices for a different bowling alley are shown in the graph
\nSo,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nSo,
\nFrom the given graph,
\nThe given line crosses the y-axis at (0, 3)
\nWe know that,
\nThe y-intercept is the value of y when the value of x is 0
\nHence, from the above,
\nWe can conclude that
\nThe given passes through (0, 3)
\nThe y-intercept is: 3<\/p>\n

Convince Me!<\/strong>
\nIn these examples, why does the y-intercept represent the cost to rent bowling shoes?
\nAnswer:
\nIn this example,
\nFrom the slope,
\nWe can determine the cost of each game
\nWe know that,
\ny-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<\/p>\n

Try It!<\/strong>
\nWhat is the y-intercept of each graph? Explain.
\n\"Envision
\nAnswer:
\nLet the given graphs be named as graph A and graph B respectively
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nSo,
\nFrom graph A,
\nThe y-intercept is: (0, 2)
\nFrom graph B,
\nThe y-intercept is: (0, -0.5)<\/p>\n

KEY CONCEPT<\/strong>
\nThe y-intercept is the y-coordinate of the point on a graph where the line crosses the y-axis.
\nWhen the line crosses through the origin, the y-intercept is 0.
\nWhen the line crosses above the origin, the y-intercept is positive.
\nWhen the line crosses below the origin, the y-intercept is negative.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> What is the y-intercept and what does it indicate?
\nAnswer:
\nThe\u00a0slope\u00a0and\u00a0y-intercept\u00a0values\u00a0indicate\u00a0characteristics of the relationship between the two variables x and\u00a0y. The\u00a0slope indicates\u00a0the rate of change in\u00a0y\u00a0per unit change in x. The\u00a0y-intercept indicates the y-value when the x-value is 0.<\/p>\n

Question 2.
\nLook for Relationships<\/strong> 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.
\nAnswer:
\nIt is given that
\nChelsea graphs a proportional relationship. Bradyn graphs a line that passes through the origin
\nSo,
\nFrom the given situation,
\nWe can observe that
\nThe 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
\nThe graph of Braydon passes through the origin i.e., the y-intercept is zero<\/p>\n

Question 3.
\nGeneralize<\/strong> When the y-intercept is positive, where does the line cross the y-axis on the graph? When it is negative?
\nAnswer:
\nWhen the y-intercept is positive, the line crosses above the origin,
\nWhen the y-intercept is negative, the line crosses below the origin<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nWhat is the y-intercept shown in the graph?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that the line passes through the origin
\nHence, from the above,
\nWe can conclude that the value of the y-intercept is: 0<\/p>\n

Question 5.
\nThe 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the relationship between the remaining time of a movie and the amount of time since Kelly hit “play.”
\nNow,
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nSo,
\nFrom the graph,
\nWe can observe that the line crosses the y-axis at (0, 1.8)
\nHence, from the above,
\nWe can conclude that the y-intercept of the graph is: 1.8
\nThe y-intercept represents the remaining time of a movie in the given situation<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nLeveled Practice Find the y-intercept of the line. The y-intercept is the point where the graph crosses the \"Envision-axis.
\nThe line crosses the y-axis at the point \"Envision
\nThe y-intercept is \"Envision
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nSo,
\nFrom the graph,
\nWe can observe that
\nThe line crosses the y-axis at the point (0, 7)
\nHence, from the above,
\nWe can conclude that
\nThe y-intercept is the point where the graph crosses the y-axis
\nThe y-intercept for the given graph is: 8<\/p>\n

Question 7.
\nFind the y-intercept of the graph.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nSo,
\nFrom the graph,
\nWe can observe that
\nThe line crosses the y-axis at the point (0, -4)
\nHence, from the above,
\nWe can conclude that
\nThe y-intercept for the given graph is: -4<\/p>\n

Question 8.
\nFind the y-intercept of the graph.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nNow,
\nFrom the graph,
\nWe can observe that
\nThe equation of the line is:
\ny = kx
\nFrom the above equation,
\nWe can say that the line passes through the origin
\nSo,
\nThe line crosses the y-axis at the point (0, 0)
\nHence, from the above,
\nWe can conclude that
\nThe y-intercept of the given graph is: 0<\/p>\n

Question 9.
\nThe 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph represents the height y, in meters, of a hot air balloon x minutes after beginning to descend
\nNow,
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nSo,
\nFrom the graph,
\nWe can observe that the line crosses the y-axis at (0, 80)
\nThe y-intercept of the graph gives us information about the height of the balloon when it began its descent
\nHence, from the above,
\nWe can conclude that the height of the balloon when it began its descent is: 80 m<\/p>\n

Question 10.
\nModel with Math<\/strong> 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?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph represents the amount of gasoline in a canister after Joshua begins to fill it at a gas station pump.
\nNow,
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nNow,
\nFrom the graph,
\nWe can observe that the line passes through the origin
\nSo,
\nThe line crosses the y-axis at the point (0, 0)
\nHence, from the above,
\nWe can conclude that
\nThe y-intercept of the given graph is: 0
\nThe y-intercept of the given graph represents the amount of gas in gallons at the starting time<\/p>\n

Question 11.
\nThe line models the temperature on a certain winter day since sunrise.
\na. What is the y-intercept of the line?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe line models the temperature on a certain winter day since sunrise.
\nNow,
\nWe know that,
\ny-coordinate of the point where the line crosses the y-axis is the y-intercept
\nSo,
\nFrom the graph,
\nWe can observe that the line crosses the y-axis at (0, 4)
\nHence, from the above,
\nWe can conclude that the y-intercept of the given line is: 4<\/p>\n

b. What does the y-intercept represent?
\nAnswer:
\nThe y-intercept of the graph gives us information about the starting temperature on a certain winter day at sunrise<\/p>\n

Question 12.
\nHigher-Order Thinking<\/strong> Your friend incorrectly makes this graph as an example of a line with a y-intercept of 3.
\na. Explain your friend’s possible error.
\n\"Envision
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the given graph,
\nWe can observe that the line crosses the y-axis at: (0, 4)
\nSo,
\nThe y-intercept of the graph is: 4
\nBut,
\nYour friend incorrectly makes this graph as an example of a line with a y-intercept of 3.
\nHence, from the above,
\nWe can conclude that the y-intercept of the given graph is 4 but not 3<\/p>\n

b. Draw a line on the graph that does represent a y-intercept of 3.
\nAnswer:
\nLet the equation with the y-intercept of 3 is:
\ny = x + 3
\nHence,
\nThe representation of the graph that does represent a y-intercept of 3 in the coordinate plane is:
\n\"\"<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 13.
\nFor each graph, draw a line through the point such that the values of the x-intercept and y-intercept are additive inverses.
\n\"Envision
\nAnswer:
\nLet the graphs be named as graph A and graph B respectively
\nNow,
\nThe given graphs are:
\n\"Envision
\nSo,
\nFrom graph A,
\nWe can observe that the x-intercept is 3 and the y-intercept is 3
\nWe know that,
\nThe “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.
\nSo,
\nThe additive inverses of the x-intercept and y-intercept are: (-3, -3)
\nFrom graph B,
\nWe can observe that the x-intercept is -3 and the y-intercept is -3
\nWe know that,
\nThe “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.
\nSo,
\nThe additive inverses of the x-intercept and y-intercept are: (3, 3)
\nHence,
\nThe representation of the additive inverses of the x and y-intercepts in the coordinate plane is:
\n\"\"<\/p>\n

Question 14.
\nWhich statements describe the graph of a proportional relationship? Select all that apply.
\n\"Envision The y-intercept is always at the point (0, 1).
\n\"Envision The line always crosses the y-axis at (0, 0).
\n\"Envision The y-intercept is 0.
\n\"Envision The y-intercept is 1.
\n\"Envision The line does NOT cross the y-axis.
\nAnswer:
\nLet the options be named as A, B, C, D, and E respectively
\nNow,
\nWe know that,
\nThe representation of the proportional relationship is:
\ny = kx
\nSo,
\nFrom the equation,
\nWe can say that the equation passes through the origin and the y-intercept is 0
\nHence, from the above,
\nWe can conclude that options B and C describes the proportional relationship<\/p>\n

Lesson 2.9 Analyze Linear Equations: y = mx + b<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explain It!<\/strong><\/p>\n

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, \u201cWe passed the one-mile mark a couple of minutes ago.”
\n\"Envision<\/p>\n

A. Construct an argument to defend Xiu’s statement.<\/p>\n

B. What mistake could Jon have made? Explain.
\nAnswer:<\/p>\n

Focus on math practices<\/strong>
\nReasoning Can you use the equation y = mx to represent the path of the tram? Is there a proportional relationship between x and y? Explain.<\/p>\n

? Essential Question<\/strong>
\nWhat is the equation of a line for a nonproportional relationship?
\nAnswer:
\nLinear equations\u00a0can be written in the form y = mx + b. When b \u2260 0, the\u00a0relationship\u00a0between x and y is\u00a0nonproportional.<\/p>\n

Try It!<\/strong>
\nWrite a linear equation in slope-intercept form for the graph shown.
\nThe y-intercept of the line is \"Envision
\n\"Envision
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that,
\nThe points are: (8, 8), and (4, 5)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nThe y-intercept of the line is the point that crosses the y-axis
\nSo,
\nFrom the given graph,
\nThe y-intercept is: 2
\nWe know that,
\nThe linear equation in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nNow,
\nm = \\(\\frac{y2<\/sub> – y1<\/sub>}{x2<\/sub> – x1<\/sub>}\\)
\n= \\(\\frac{5 – 8}{4 – 8}\\)
\n= \\(\\frac{3}{4}\\)
\nSo,
\nThe linear equation in the slope-intercept form is:
\ny = \\(\\frac{3}{4}\\)x + 2
\ny = \\(\\frac{3x + 8}{4}\\)
\n4y = 3x + 8
\nHence, from the above,
\nWe can conclude that the linear equation in the slope-intercept form is:
\n4y = 3x + 8<\/p>\n

Convince Me!<\/strong><\/p>\n

What two values do you need to know to write an equation of a line, and how are they used to represent a line?
\nAnswer:
\nTo write an equation of a line in the slope-intercept form,
\nThe two values you need to know are:
\nA) Slope of a line and it is represented as “m”
\nB) The y-intercept of a line and is represented as “c”<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

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.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> What is the equation of a line for a nonproportional relationship?
\nAnswer:
\nLinear equations\u00a0can be written in the form y = mx + b. When b \u2260 0, the\u00a0relationship\u00a0between x and y is\u00a0nonproportional.<\/p>\n

Question 2.
\nUse Structure<\/strong> 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?
\nAnswer:
\nIt is given that
\nThe donations by a restaurant to a certain charity, y, will be two-fifths of its profits, x, plus $50.
\nSo,
\nDonations to a certain charity by a restaurant = The part of the profits of a restaurant + $50
\ny = \\(\\frac{2}{5}\\)x + $50
\nCompare the above equation with
\ny = mx + c
\nWhere,
\nm is the slope of a line
\nc is the y-intercept of a line
\nSo,
\nWhen we compare the equation,
\nThe slope of a line is (m): \\(\\frac{2}{5}\\)
\nThe y-intercept of a line is (c) : $50<\/p>\n

Question 3.
\nBe Precise<\/strong> 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.
\nAnswer:
\nIt is given that
\nPriya 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.
\nNow,
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm is the slope of the line
\nc is the y-intercept
\nSo,
\nBy comparing,
\nWe get,
\nm = \\(\\frac{3}{4}\\)
\nc = -4
\nNow,
\nFrom the y-intercept,
\nWe can say that the y-intercept lies below the origin i.e., in the 3rd quadrant
\nFrom the slope of the line,
\nWe can say that the value of m lies in the 1st quadrant
\nHence, from the above,
\nWe can conclude that the line drawn by Priya will be in the 4th quadrant for the above values of c and m<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nChrissie 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nChrissie 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.
\nNow,
\nThe given graph is
\n\"Envision
\nFrom the given graph,
\nThe y-intercept is: 5
\nNow,
\nWhen we observe the given two equations,
\nThe slope is the same and the y-intercepts are different and the correct y-intercept must be 5
\nHence, from the above,
\nWe can conclude that George is correct<\/p>\n

Question 5.
\nFara 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.
\na. 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nFara 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.
\nNow,
\nThe total cost of the rent = The cost of the rent per hour + Additional set-up fee
\nSo,
\ny = 500x + 100
\nHence,
\nThe representation of the above equation in the coordinate plane is:
\n\"\"<\/p>\n

b. What is the equation of the line in slope-intercept form?
\nAnswer:
\nWe know that,
\nThe total cost of the rent = The cost of the rent per hour + Additional set-up fee
\nSo,
\ny = 500x + 100
\nWhere,
\nx is the number of hours
\nThe above equation is in the form of
\ny = mx + c
\nWhich is the slope-intercept form of the equation
\nHence, from the above,
\nWe can conclude that the equation of the line in the slope-intercept form is:
\ny = 500x + 100<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nLeveled Practice What is the graph of the equation y = 2x + 4?
\nThe y-intercept is \"Envision, which means the line crosses the y-axis at the point (\"Envision). Plot this point.
\nThe slope of the line is positive, so it goes \"Envision from left to right.
\nStart at the y-intercept. Move up \"Envision, and then move right \"Envision
\nYou are now at the point (\"Envision). Plot this point. Draw a line to connect the two points.
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = 2x + 4
\nSo,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"
\nCompare the given equation with
\ny = mx + c
\nWher,
\nm is the slope of a line
\nc is the y-intercept of a line
\nSo,
\nThe y-intercept of the given graph is 4 which means the line crosses the y-axis at the point (0, 4)
\nThe slope of the line is positive, so it goes up from left to right.
\nStart at the y-intercept. Move up 2 units, and then move right 2 units
\nSo,
\nYou are now at the point (3, 10).<\/p>\n

Question 7.
\nWrite an equation for the line in slope-intercept form.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nNow,
\nFrom the given graph,
\nWe can observe that the y-intercept is: -3
\nNow,
\nThe given points from the graph to find the slope are: (-2, -2), and (4, -5)
\nNow,
\nSLope (m) = \\(\\frac{-5\u00a0 – (-2)}{4 – (-2)}\\)
\nm = \\(\\frac{-3}{6}\\)
\nm = –\\(\\frac{1}{2}\\)
\nSo,
\nThe equation of the line in the slope-intercept form is:
\ny = –\\(\\frac{1}{2}\\)x – 3
\nHence, from the above,
\nWe can conclude that the equation of the line in the slope-intercept form is:
\ny = –\\(\\frac{1}{2}\\)x – 3<\/p>\n

Question 8.
\nWrite an equation for the line in slope-intercept form.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nNow,
\nFrom the given graph,
\nWe can observe that the y-intercept is: 4
\nNow,
\nThe given points from the graph to find the slope are: (1, 1), and (0, 4)
\nNow,
\nSLope (m) = \\(\\frac{4 – 1}{0 – 1}\\)
\nm = \\(\\frac{3}{-1}\\)
\nm = -3
\nSo,
\nThe equation of the line in the slope-intercept form is:
\ny = -3x + 4
\nHence, from the above,
\nWe can conclude that the equation of the line in the slope-intercept form is:
\ny = -3x + 4<\/p>\n

Question 9.
\nThe 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe line models the cost of renting a kayak
\nwhere,
\nx is the number of hours the kayak is rented and y is the total cost of renting the kayak.
\nNw,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nThe y-intercept of the graph is: 5
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nNow,
\nFrom the given graph,
\nThe points to find the graph are: (3, 40), and (2, 30)
\nSo,
\nSlope (m) = \\(\\frac{30 – 40}{2 – 3}\\)
\n= 10
\nSo,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = 10x + 5
\nHence, from the above,
\nWe can conclude that
\nThe equation of the line in the slope-intercept form is:
\ny = 10x + 5<\/p>\n

Question 10.
\nGraph the equation y = 3x – 5.
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = 3x – 5
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

Question 11.
\nAmy 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.
\na. What is the correct equation for the line in slope-intercept form?
\n\"Envision
\nAnswer:
\nIt is given that
\nAmy began with $25 in her bank account and spent $5 each day. The line shows the amount of money in her bank account
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nThe y-intercept is: 25
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nNow,
\nThe given points to find the slope are: (5, 0), and (1, 20)
\nSo,
\nSlope (m) = \\(\\frac{20 – 0}{5 – 1}\\)
\n= \\(\\frac{20}{5}\\)
\n= 4
\nSo,
\nThe equation of the line in the slope-intercept form is:
\ny = 4x + 25
\nHence, from the above,
\nWe can conclude that the equation of the line in the slope-intercept form is:
\ny = 4x + 25<\/p>\n

b. Critique Reasoning What mistake might Amy have made?
\nAnswer:
\nAnswer:
\nThe mistakes might made by Amy are:
\nA) The value of y-intercept is 25 and the value of x-intercept is: 5
\nB) The slope is not negative as it moves down from top to bottom<\/p>\n

Question 12.
\nHigher-Order Thinking<\/strong> The line represents the cost of ordering concert tickets online.
\na. Write an equation for the line in slope-intercept form, where x is the number of tickets and y is the total cost.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe line represents the cost of ordering concert tickets online.
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nThe y-intercept is: 10
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nNow,
\nThe given points to find the slope are: (1, 33.25), and (0, 12.25)
\nSo,
\nSlope (m) = \\(\\frac{12.25 – 33.25}{0 – 1}\\)
\n= \\(\\frac{21}{1}\\)
\n= 21
\nSo,
\nThe equation of the line in the slope-intercept form is:
\ny = 21x + 10
\nHence, from the above,
\nWe can conclude that the equation of the line in the slope-intercept form is:
\ny = 21x + 10<\/p>\n

b. Explain how you can write an equation for this situation without using a graph.
\nAnswer:
\nWe know that,
\nThe total cost of ordering concert tickets online = (The cost of 1 Ticket) \u00d7 (The number of Tickets) + Processing fee
\nLet the number of tickets be x
\nLet the total cost of ordering concert tickets online be y
\nSo,
\ny = 21x + 10
\nHence, from the above,
\nWe can conclude that the equation for this situation without using a graph is:
\ny = 21x + 10<\/p>\n

c. Is this graph a good representation of the situation? Explain.
\nAnswer:
\nYes,
\nThe 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<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 13.
\nWhat should you do first to graph the equation y = \\(\\frac{2}{5}\\)x – 1?
\nA. Plot the point (0, 0).
\nB. Plot the point (2, 5).
\nC. Plot a point at the x-intercept.
\nD. Plot a point at the y-intercept.
\nAnswer:
\nThe given equation is:
\ny = \\(\\frac{2}{5}\\)x – 1
\nCompare the above equation with
\ny = mx + c
\nHence, from the above,
\nWe can conclude that the first step to draw the graph for the given equation is:
\nPlot a point at the y-intercept<\/p>\n

Question 14.
\nWrite an equation for the line in slope-intercept form.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nThe y-intercept of the graph is: 8
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nNow,
\nFrom the given graph,
\nThe points to find the graph are: (4, 0), and (0, 8)
\nSo,
\nSlope (m) = \\(\\frac{8 – 0}{0 – 4}\\)
\n= -2
\nSo,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = -2x + 8
\nHence, from the above,
\nWe can conclude that
\nThe equation of the line in the slope-intercept form is:
\ny = -2x + 8<\/p>\n

TOPIC 2 REVIEW<\/h3>\n

? Topic Essential Question<\/strong><\/p>\n

How can you analyze connections between linear equations and use them to solve problems?
\nAnswer:
\nAssuming 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\u00a0use\u00a0the resulting value in the other<\/p>\n

Vocabulary Review<\/strong><\/p>\n

Complete each definition and provide an example of each vocabulary word.<\/p>\n

\"Envision<\/p>\n

\"Envision<\/p>\n

Question 1.
\nThe change in y divided by the change in x is the ____
\nAnswer:
\nThe change in y divided by the change in x is defined as the “Slope of a line”
\nExample:
\nSlope = \\(\\frac{y}{x}\\)
\n= \\(\\frac{2}{5}\\)<\/p>\n

Question 2.
\nThe point on the graph where the line crosses the y-axis is the ____ of a line.
\nAnswer:
\nThe 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
\nExample:
\nThe point on the graph where the line crosses the y-axis is at (0, 2)
\nSo,
\nThe y-intercept is: 2<\/p>\n

Question 3.
\nThe ____ 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 ___
\nAnswer:
\nThe “Slope-intercept form” of a line is
\ny = mx + b
\nThe variable m in the equation stands for the x-intercept.
\nThe variable b in the equation stands for the y-intercept<\/p>\n

Use Vocabulary in Writing<\/strong>
\nPaddle 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.
\nAnswer:
\nIt is given that
\nPaddleboats rent for a fee of $25, plus an additional $12 per hour.
\nWhere,
\nx represents the cost to rent a paddleboat for x hours
\nNow,
\nThe total cost to rent a paddleboat = The cost of a paddleboat per hour + $12
\ny = $25x + $12
\nHence, from the above,
\nWe can conclude that the equation of the line for this situation is:
\ny = $25x + $12<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

LESSON 2.1 Combine Like Terms to Solve Equations<\/strong><\/p>\n

Quick Review<\/strong>
\nYou 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.<\/p>\n

Practice<\/strong>
\nSolve each equation for x.<\/p>\n

Question 1.
\n2x + 6x = 1,000
\nAnswer:
\nThe given equation is:
\n2x + 6x = 1,000
\nSo,
\n8x = 1,000
\nDivide by 8 into both sides
\nx = \\(\\frac{1,000}{8}\\)
\nx = 125
\nHence, from the above,
\nWe can conclude that the value of x is: 125<\/p>\n

Question 2.
\n2\\(\\frac{1}{4}\\)x + 2\\(\\frac{1}{2}\\)x = 44
\nAnswer:
\nThe given equation is:
\n2\\(\\frac{1}{4}\\)x + 2\\(\\frac{1}{2}\\)x = 44
\nWe know that,
\n2\\(\\frac{1}{4}\\) = \\(\\frac{9}{4}\\)
\n2\\(\\frac{1}{2}\\) = \\(\\frac{5}{2}\\)
\nSo,
\n\\(\\frac{9}{4}\\)x + \\(\\frac{5}{2}\\)x = 44
\n\\(\\frac{19}{4}\\)x = 44
\nMultiply with \\(\\frac{4}{19}\\) on both sides
\nSo,
\nx = 44 \u00d7 \\(\\frac{4}{19}\\)
\nx = \\(\\frac{88}{19}\\)
\nHence, from the above,
\nWe can conclude that the value of x is: \\(\\frac{88}{19}\\)<\/p>\n

Question 3.
\n-2.3x – 4.2x = -66.3
\nAnswer:
\nThe given equation is:
\n-2.3x – 4.2x = -66.3
\nSo,
\n-6.5x = -66.3
\n6.5x = 66.3
\nDivide by 6.5 into both sides
\nSo,
\nx = \\(\\frac{66.3}{6.5}\\)
\nx = \\(\\frac{51}{5}\\)
\nx = 10.2
\nHence, from the above,
\nWe can conclude that the value of x is: 10.2<\/p>\n

Question 4.
\nJavier bought a microwave for $105. The cost was 30% off the original price. What was the price of the microwave before the sale?
\nAnswer:
\nIt is given that
\nJavier bought a microwave for $105. The cost was 30% off the original price
\nSo,
\nThe price of the microwave before the sale = The price of the microwave + 30% of the price of the microwave
\n= $105 + \\(\\frac{30}{100}\\) ($105)
\n= $105 (\\(\\frac{130}{100}\\))
\n= \\(\\frac{13650}{100}\\)
\n= $136.5
\nHence, from the above,
\nWe can conclude that the price of the microwave before the sale is: $136.5<\/p>\n

LESSON 2.2 Solve Equations with Variables on Both Sides<\/strong><\/p>\n

Quick Review<\/strong>
\nIf 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.<\/p>\n

Practice<\/strong>
\nSolve each equation for x.<\/p>\n

Question 1.
\n3x + 9x = 6x + 42
\nAnswer:
\nThe given equation is:
\n3x + 9x = 6x + 42
\n12x = 6x + 42
\nRearrange the like terms
\nSo,
\n12x – 6x = 42
\n6x = 42
\nSo,
\nx = \\(\\frac{42}{6}\\)
\nx = 7
\nHence, from the above,
\nWe can conclude that the value of x is: 7<\/p>\n

Question 2.
\n\\(\\frac{4}{3}\\)x + \\(\\frac{2}{3}\\)x = \\(\\frac{1}{3}\\)x + 5
\nAnswer:
\nThe given equation is:
\n\\(\\frac{4}{3}\\)x + \\(\\frac{2}{3}\\)x = \\(\\frac{1}{3}\\)x + 5
\nSo,
\n\\(\\frac{6}{3}\\)x = \\(\\frac{1}{3}\\)x + 5
\n\\(\\frac{6}{3}\\)x – \\(\\frac{1}{3}[latex]x = 5
\n[latex]\\frac{5}{3}\\)x = 5
\nMultiply with \\(\\frac{3}{5}\\) on both sides
\nSo,
\nx = 5 \u00d7 \\(\\frac{3}{5}\\)
\nx = 3
\nHence, from the above,
\nWe can conclude that the value of x is: 3<\/p>\n

Question 3.
\n9x \u2013 5x + 18 = 2x + 34
\nAnswer:
\nThe given equation is:
\n9x – 5x + 18 = 2x + 34
\nSo,
\n4x + 18 = 2x + 34
\nRearrange the like terms
\nSo,
\n4x – 2x = 34 – 18
\n2x = 16
\nDivide by 2 into both sides
\nSo,
\nx = \\(\\frac{16}{2}\\)
\nx = 8
\nHence, from the above,
\nWe can conclude that the value of x is: 8<\/p>\n

Question 4.
\nMegan 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?
\nAnswer:
\nIt is given that
\nMegan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week.
\nNow,
\nLet x be the number of weeks
\nSo,
\nThe money saved by Megan = $50 + $5.50x
\nThe money saved by Connor = $18.50 + $7.75x
\nSo,
\nTo find out after how many weeks Megan and Connor have saved the same amount,
\n$50 + $5.50x = $18.50 + $7.75x
\nRearrange the like terms
\nSo,
\n$50 – $18.50 = $7.75x – $5.50x
\n$31.05 = $2.25x
\nDivide by 2.25 into both sides
\nSo,
\nx = \\(\\frac{31.05}{2.25}\\)
\nx = 13.8
\nx = 14 weeks 1 day
\nx \u2245 14 weeks
\nHence, from the above,
\nWe can conclude that after approximately 14 weeks, Megan and Connor have saved the same amount<\/p>\n

LESSON 2.3 Solve Multistep Equations<\/strong><\/p>\n

Quick Review<\/strong>
\nWhen 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.<\/p>\n

Practice Solve each equation for x.<\/p>\n

Question 1.
\n4(x + 4) + 2x = 52
\nAnswer:
\nThe given equation is:
\n4 (x + 4) + 2x = 52
\nSo,
\n4 (x) + 4 (4) + 2x = 52
\n4x + 16 + 2x = 52
\n6x + 16 = 52
\nRearrange the like terms
\nSo,
\n6x = 52 – 16
\n6x = 36
\nx = \\(\\frac{36}{6}\\)
\nx = 6
\nHence, from the above,
\nWe can conclude that the value of x is: 6<\/p>\n

Question 2.
\n8(2x + 3x + 2) = -4x + 148
\nAnswer:
\nThe given equation is:
\n8 (2x + 3x + 2) = -4x + 148
\nSo,
\n8 (5x + 2) = -4x + 148
\n8 (5x) + 8 (2) = -4x + 148
\n40x + 16 = -4x + 148
\nRearrange the like terms
\nSo,
\n40x + 4x = 148 – 16
\n44x = 132
\nx = \\(\\frac{132}{4}\\)
\nx = 3
\nHence, from the above,
\nWe can conclude that the value of x is: 3<\/p>\n

Question 3.
\nJustin 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?
\nAnswer:
\nIt is given that
\nJustin 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.
\nSo,
\nTotal spent money of Justin = The original price of binder + The original price of a calculator
\nLet the original price of the calculator be x
\nSo,
\n$6.20 + 30% of $6.20 + x + 30% of x = $107.27
\n$6.20 + \\(\\frac{3}{10}\\) ($6.20) + x + \\(\\frac{3}{10}\\) of x = $107.27
\n$6.20 + 1.86 + 1.3x = $107.27
\n$8.06 + 1.3x = $107.27
\n1.3x = $107.27 – $8.06
\n1.3x = 99.21
\nx = \\(\\frac{99.21}{1.3}\\)
\nx = 76.31
\nHence, from the above,
\nWe can conclude that the original price of the calculator is: $76.31<\/p>\n

LESSON 2.4 Equations with No Solutions or Infinitely Many Solutions<\/strong><\/p>\n

Quick Review<\/strong>
\nWhen 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.<\/p>\n

Practice<\/strong>
\nHow many solutions does each equation have?<\/p>\n

Question 1.
\nx + 5.5 + 8 = 5x \u2013 13.5 – 4x
\nAnswer:
\nThe given equation is:
\nx + 5.5 + 8 = 5x – 13.5 – 4x
\nSo,
\nx + 13.5 = x – 13.5
\nSubtract with x on both sides
\nSo,
\n13.5 = -13.5
\nHence, from the above,
\nwe can conclude that there are no solutions for the given equation<\/p>\n

Question 2.
\n4(\\(\\frac{1}{2}\\)x + 3) = 3x + 12 – x
\nAnswer:
\nThe given equation is:
\n4(\\(\\frac{1}{2}\\)x + 3) = 3x + 12 – x
\nSo,
\n4 \u00d7 \\(\\frac{1}{2}\\)x + 4 (3) = 3x + 12 – x
\n2x + 12 = 2x + 12
\nSubtract with 2x on both sides
\nSo,
\n12 = 12
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given equation<\/p>\n

Question 3.
\n2(6x + 9 – 3x) = 5x + 21
\nAnswer:
\nThe given equation is:
\n2 (6x + 9 – 3x) = 5x + 21
\nSo,
\n2 (3x + 9) = 5x + 21
\n2 (3x) + 2 (9) = 5x + 21
\n6x + 18 = 5x + 21
\nRearrange the like terms
\nSo,
\n6x – 5x = 21 – 18
\nx = 3
\nHence, from the above,
\nWe can conclude that there is only 1 solution for the given equation<\/p>\n

Question 4.
\nThe 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.
\nAnswer:
\nIt is given that
\nThe 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.
\nNow,
\nTo find out whether the weight of the dogs will be the same or not,
\n2 (2x + 3) = 3 (3x + 1)
\nSo,
\n2 (2x) + 2 (3) = 3 (3x) + 3 (1)
\n4x + 6 = 9x + 3
\nRearrange the like terms
\nSo,
\n9x – 4x = 6 – 3
\n5x = 3
\nx = \\(\\frac{3}{5}\\)
\nSo,
\nThere is only 1 solution for the given equation
\nHence, from the above,
\nWe can conclude that the weights of the dogs will be the same<\/p>\n

LESSON 2.5 Compare Proportional Relationships<\/strong><\/p>\n

Quick Review<\/strong>
\nTo compare proportional relationships, compare the rate of change or find the unit rate.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nTwo trains are traveling at a constant rate. Find the rate of each train. Which train is traveling at the faster rate?
\n\"Envision
\n\"Envision
\nAnswer:
\nWe know that,
\nUnit rate = \\(\\frac{y}{x}\\)
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nNow,
\nFor Train A,
\nUnit rate = \\(\\frac{A value of Distance}{The value of time that corresponds to the Distance}\\)
\n= \\(\\frac{50}{2}\\)
\n= 25 miles per hour
\nFor Train B,
\nUnit rate = \\(\\frac{y}{x}\\)
\n= \\(\\frac{20}{1}\\)
\n= 20 miles per hour
\nSo,
\nUnit rate of Train A > Unit rate of Train B
\nHence, from the above,
\nWe can conclude that Train A is the fastest<\/p>\n

Question 2.
\nA 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?
\nAnswer:
\nIt is given that
\nA 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.
\nSo,
\nThe cost per ounce of water of store A = \\(\\frac{The cost of a 16-ounce bottle of water}{16}\\)
\n= \\(\\frac{$1.28}{16}\\)
\n= $0.08
\nThe cost per ounce of water of store B = \\(\\frac{y}{x}\\)
\n= $0.07
\nSo,
\nThe cost per ounce of water of store A > The cost per ounce of water of store B
\nHence, from the above,
\nWe can conclude that the cost per ounce of water of store B costs less per ounce<\/p>\n

LESSON 2.6 Connect Proportional Relationships and Slope<\/strong><\/p>\n

Quick Review<\/strong>
\nThe slope of a line in a proportional relationship is the same as the unit rate and the constant of proportionality.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nThe 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.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the proportions of blue paint and yellow paint that Briana mixes to make green paint.
\nNow,
\nThe given graph is:
\n\"Envision
\nSo,
\nFrom the graph,
\nThe slope of the given line = \\(\\frac{y}{x}\\)
\n= \\(\\frac{5}{6}\\)
\nHence, from the above slope of the line,
\nWe can conclude that for 5 parts of yellow paint, we have to mix 6 parts of blue paint to make green paint<\/p>\n

LESSON 2.7 Analyze Linear Equations: y = mx<\/strong><\/p>\n

Quick Review<\/strong>
\nA proportional relationship can be represented by an equation in the form y = mx, where m is the slope.<\/p>\n

Practice<\/strong>
\nA mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.<\/p>\n

Question 1.
\nWrite a linear equation that represents the relationship between peanuts, x, and walnuts, y.
\nAnswer:
\nIt is given that
\nA mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.
\nWe know that,
\nSlope (m) = \\(\\frac{y}{x}\\)
\nm = \\(\\frac{1}{3}\\)
\nWe know that,
\nThe linear equation that represents the relationship between peanuts and walnuts is:
\ny = mx
\nSo,
\ny = \\(\\frac{1}{3}\\)x
\nx = 3y
\nHence, from the above,
\nWe can conclude that the linear equation that represents the relationship between peanuts and walnuts is:
\nx = 3y<\/p>\n

Question 2.
\nGraph the line.
\n\"Envision
\nAnswer:
\nThe linear equation that represents the relationship between peanuts and walnuts is:
\nx = 3y
\nHence,
\nThe representation of the linear equation in the coordinate plane is:
\n\"\"<\/p>\n

LESSON 2.8 Understand the y-Intercept of a Line<\/strong><\/p>\n

Quick Review<\/strong>
\nThe 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.<\/p>\n

Practice<\/strong>
\nThe equation y = 5 +0.5x represents the cost of getting a car wash and using the vacuum for x minutes.
\n\"Envision<\/p>\n

Question 1.
\nWhat is the y-intercept?
\nAnswer:
\nWe know that,
\nThe equation of the line in the y-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nNow,
\nThe given equation is:
\ny = 5 + 0.5x
\nHence, from the above,
\nWe can conclude that the y-intercept is: 5<\/p>\n

Question 2.
\nWhat does the y-intercept represent?
\nAnswer:
\nThe y-intercept in the given situation represents that the initial cost of getting a car wash using the Vaccum<\/p>\n

LESSON 2.9 Analyze Linear Equations: y = mx + b<\/strong><\/p>\n

Quick Review<\/strong>
\nAn 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.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nGraph the line with the equation y = \\(\\frac{1}{2}\\)x – 1.
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = \\(\\frac{1}{2}\\)x – 1
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

Question 2.
\nWhat is the equation of the line?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nThe y-intercept is: 3
\nNow,
\nWe know that,
\nThe equation of the line in the slope-intercept form is:
\ny = mx + c
\nNow,
\nTo find the slope,
\nThe points are: (0, 3), and (3, 0)
\nSo,
\nSlope (m) = \\(\\frac{0 – 3}{3 – 0}\\)
\n= \\(\\frac{-3}{3}\\)
\n= -1
\nHence, from the above,
\nWe can conclude that the equation of the line in the slope-intercept form is:
\ny = -x + 3<\/p>\n

Topic 2 Fluency Practice<\/h3>\n

Pathfinder<\/strong><\/p>\n

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.<\/p>\n

\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

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