Category: Envision Math

Go through the enVision Math Common Core Grade 1 Answer Key Topic 1 Understand Addition and Subtraction regularly and improve your accuracy in solving questions.

enVision Math Common Core 1st Grade Answers Key Topic 1 Understand Addition and Subtraction

Essential Question:

What are ways to think about addition and subtraction?

Look at the adult and baby giraffes.

How are they the same? How are they different?

Wow! Let’s do this project and learn more.

envision STEM Project: Parents and Babies

Find Out Talk to friends and relatives about different animals and their babies.
Ask them how the parents and babies are the same and how they are different.

Journal: Make a Book Show what you found.
In your book, also:

• Draw some animals, including the parents and babies.
• Create and solve addition and subtraction stories about some animals and their babies.

Review What You Know
Vocabulary

Question 1.
Count the fish. Write the number that tells how many.

Answer:
Number of fishes given in the picture = 5 or Five.

Explanation:
Number of fishes given in the picture = 5 or Five.

 

Question 2.
Join the two groups and write how many.

Answer:
Total number of apples in the figure = 4 or Four.

Explanation:
Number of black apples in the figure = 2 or Two.
Number of grey apples in the figure = 2 or Two.
Total number of apples in the figure = Number of black apples in the figure + Number of grey apples in the figure
= 2 + 2
= 4 or Four.

 

Question 3.
Write how many soccer balls there are in all.

Answer:
Total number of balls in the figure = 4 or Four.

Explanation:
Number of foot balls in the figure = 3 or Three.
Number of tennis balls in the figure =1 or One.
Total number of balls in the figure = Number of foot balls in the figure + Number of tennis balls in the figure
= 3 + 1
= 4 or Four.

 

Counting
Question 4.
Tammy has 4 balloons. Draw a picture of her balloons.
Answer:
Number of balloons Tammy has = 4 or Four.

Explanation:
Number of balloons Tammy has = 4 or Four.

 

Question 5.
Write the number that tells how many cats.

Answer:
Number of cats in the given picture = 3 or Three.

Explanation:
Number of cats in the given picture = 3 or Three.

 

Sums
Question 6.
Circle the number that shows how many crabs you see.

Answer:
Number of crabs in the given picture = 3 or Three.

Explanation:
Number of crabs in the given picture = 3 or Three.
Number of scorpions in the given picture = 2 or Two.

 

Pick a Project
Project 1A

Where do birds lay their eggs?
Project: Draw a bird Clutch

Answer:
Birds lay their Eggs in their nests.

Explanation:

 

Project 1B
What is the most popular fruit juice in the world?
Project: Find Fruit Facts

Answer:
The most popular fruit juice in the world  is orange juice.

Explanation:
The most popular fruit juice in the world  is orange juice.

Project 1C
What are different homes made of?
Project: Build a model

Answer:
Many houses are now made with steel frames put together with rivets and bolts.

Explanation:
Generally houses are made of steel , iron, mud ,clay ,stones , cement.

Lesson 1.1 Add To
Solve and Share

4 dogs
Some dogs join.
How many dogs now?
Show how you solve.

_______ dogs now
Answer:
Total number of dogs = 8 or Eight.
___8____ dogs now.

Explanation:
Number of dogs given in the picture = 4 or Four.
Number of dogs joined more = 4 or Four.
Total number of dogs = Number of dogs given in the picture  + Number of dogs joined more
= 4 + 4
= 8 or Eight.

Visual Learning Bridge

Convince Me!
How do cubes help you solve the problem?
Answer:
Cubes helps to solve the problem easier in counting.

Explanation:
Cubes helps to solve the problem easier in counting without any confusion.

 

 

Guided Practice
Solve. Use cubes to help.

Question 1.
3 cows

3 cows join.

How many cows now?

Answer:
Total number of cows now = 6 or Six.

Explanation:
Number of cows in the given picture = 3 or Three.
Number of cows more joined = 3 or Three.
Total number of cows now = Number of cows in the given picture + Number of cows more joined
= 3 + 3
= 6 or Six.
   

 

Question 2.
2 birds

6 birds join.

How many birds now?

Answer:
Total number of birds now = 8 or Eight.

Explanation:
Number of birds in the given picture = 2 or Two.
Number of birds more joined = 6 or Six.
Total number of birds now = Number of birds in the given picture + Number of birds more joined
= 2 + 6
= 8 or Eight.

 

 

Question 3.
4 bees

4 bees join

How many bees now?

Answer:
Total number of bees now = 8 or Eight.

Explanation:
Number of bees in the given picture = 4 or Four.
Number of bees more joined = 4 or Four.
Total number of bees now = Number of bees in the given picture + Number of bees more joined
= 4 + 4
= 8 or Eight.

 

Question 4.
3 bugs

6 bugs join.

How many bugs now?

Answer:
Total number of bugs now = 9 or Nine.

Explanation:
Number of bugs in the given picture = 3 or Three.
Number of bugs more joined = 6 or Six.
Total number of bugs now = Number of bugs in the given picture + Number of bugs more joined
= 3 + 6
= 9 or Nine.

 

Solve the problem.
Draw a picture to help.

Question 5.
Higher Order Thinking
6 ducks
4 chickens
2 ducks join.
How many ducks in all?
_____ ducks in all.
Answer:
Total number of ducks now in all = 8 or Eight.
__8___ ducks in all.

Explanation:
Number of ducks  given = 6 or Six.
Number of chickens given = 4 or Four.
Number of ducks more joined = 2 or Two.
Total number of ducks now in all = Number of ducks given + Number of ducks more joined
= 6 + 2
= 8 or Eight.

1

Problem Solving
Solve each problem.

Question 6.
Vocabulary
3 dogs
4 dogs join.
Add to find the sum.

Answer:
Total number of dogs now in all = 7 or Seven.

Explanation:
Number of dogs given = 3 or Three.
Number of dogs more joined = 4 or Four.
Total number of dogs now in all = Number of dogs given + Number of dogs more joined
= 3 + 4
= 7 or Seven.

 

Question 7.
Model
8 cats
1 cat joins.
How many cats now?

Answer:
Total number of cats now in all = 9 or Nine.

Explanation:
Number of cats given in the picture = 8 or Eight.
Number of cats more joined = 1 or One.
Total number of cats now in all = Number of cats given in the picture + Number of cats more joined
= 8 + 1
= 9 or Nine.

 

Question 8.
Higher Order Thinking
Make up an addition story about the birds.

Answer:
There are 3 birds on a tree sitting. Later 1 bird joined them. How many birds are in all.
Total number of birds in all now = 4 or Four.

Explanation:
Number of birds sitting on tree = 3 or Three.
Number of birds joined later = 1 or One.
Total number of birds in all now = Number of birds sitting on tree + Number of birds joined later
= 3 + 1
= 4 or Four.

 

 

Question 9.
Assessment Practice

5 frogs

3 frogs join

How many frogs now?
A. 5 + 1 = 6 frogs
B. 5 + 2 = 7 frogs
C. 5 + 3 = 8 frogs
D. 5 + 4 = 9 frogs
Answer:
Total number of frogs now in all = 8 or Eight.
C. 5 + 3 = 8 frogs.

Explanation:
Number of frogs given = 5 or Five.
Number of frogs more joined = 3 or Three.
Total number of frogs now in all = Number of frogs given + Number of frogs more joined
= 5 + 3
= 8 or Eight.

 

 

Lesson 1.2 Put Together

4 red apples and 4 green apples How many apples in all?
Show how you solve.
Use cubes to help.

___ apples in all
Answer:
Total number of apples now in all = 8 or Eight.
_8__ apples in all.

Explanation:
Number of red apples = 4 or Four.
Number of green apples = 4 or Four.
Total number of apples now in all = Number of red apples + Number of green apples
= 4 + 4
= 8 or Eight.

 

Visual Learning Bridge

Convince Me!

Use cubes.
show 4 + 2.
Then show 2 + 4.
What do you notice?
Answer:
Sum of 4 + 2 = ??
4 + 2 = 6 or Six.
Sum of 2 + 4 = ??
2 + 4 = 6 or Six.

It is noticed the sum of (4 + 2) is same as (2 + 4) = 6 or Six.

Explanation:
Sum of 4 + 2 = ??
4 + 2 = 6 or Six.

Sum of 2 + 4 = ??
2 + 4 = 6 or Six.

 

Guided Practice
Solve. Use cubes to help. Write an addition equation.

Question 1.
3 yellow birds and 5 blue birds

How many birds in all?

Answer:
Total number of birds in all = 8 or Eight.

Explanation:
Number of yellow birds = 3 or Three.
Number of blue birds = 5 or Five.
Total number of birds in all = Number of yellow birds + Number of blue birds
= 3 + 5
= 8 or Eight.

 

Question 2.
1 white egg and 6 blue eggs

How many eggs in all?

Answer:
Total number of eggs in all = 7 or Seven.

Explanation:
Number of white eggs = 1 or One.
Number of blue eggs = 6 or Six.
Total number of eggs in all = Number of white eggs + Number of blue eggs
= 1 + 6
= 7 or Seven.

Independent Practice
Solve. Use cubes or draw a picture. Write an addition equation.

Question 3.
3 little pigs and 4 big pigs

How many pigs in all?

Answer:
Total number of pigs in all = 7 or Seven.

Explanation:
Number of little pigs = 3 or Three.
Number of big pigs = 4 or Four.
Total number of pigs in all = Number of little pigs + Number of big pigs
= 3 + 4
= 7 or Seven.

 

 

Question 4.
3 box cars and 3 tank cars

How many cars in all?

Answer:
Total number of cars in all = 6 or Six.

Explanation:
Number of box cars = 3 or Three.
Number of tank cars = 3 or Three.
Total number of cars in all = Number of box cars + Number of tank cars
= 3 + 3
= 6 or Six.

 

 

 

Question 5.
Higher Order Thinking
2 red hats
3 shoes
7 blue hats
How many hats in all?
Draw a picture.
Write an addition equation.

Answer:
Total number of hats in all = 9 or Nine.

Explanation:
Number of red hats = 2 or Two.
Number of shoes = 3 or Three.
Number of blue hats = 7 or Seven.
Total number of hats in all = Number of red hats + Number of blue hats
= 2 + 7
= 9 or Nine.

 

Problem Solving
Solve each problem.
Question 6.
Make Sense
Jen has 2 red flowers and 5 blue flowers. How many flowers in all? Write an equation.

Answer:
Total number of flowers Jen has in all = 7 or Seven.

Explanation:
Number of red flowers Jen has = 2 or Two.
Number of blue flowers Jen has = 5 or Five.
Total number of flowers Jen has in all = Number of red flowers Jen has + Number of blue flowers Jen has
= 2 + 5
= 7 or Seven.

 

 

Question 7.
Higher Order Thinking
Write a picture story. Show blue fish and green fish. Write an addition equation. Tell how many fish in all.

Answer:
There are 2 blue fishes in a tank. 2 green fishes were added into the tank. How many fishes are all in the tank.
Total number of fishes all in the tank = 4 or Four.

Explanation:
There are 2 blue fishes in a tank. 2 green fishes were added into the tank. How many fishes are all in the tank.
Number of blue fishes in the tank = 2 or Two.
Number of green fishes in the tank = 2 or Two.
Total number of fishes all in the tank = Number of blue fishes in the tank + Number of green fishes in the tank
= 2 + 2
= 4 or Four.

 

Question 8.
Assessment Practice
4 red apples and 5 green apples

How many apples in all?

Answer:
Total number of  apples in all = 9 or Nine.
B. 4 + 5 = 9 apples.

Explanation:
Number of red apples = 4 or Four.
Number of green apples = 5 or Five.
Total number of  apples in all = Number of red apples + Number of green apples
= 4 + 5
= 9 or Nine.

 

 

 

Lesson 1.3 Both Addends Unknown

Sarah has 5 pencils.
She puts some in the green cup.
She puts some in the red cup.
How many pencils could be in each cup?

Answer:
Number of pencils Sarah puts in green cup = 3 or Three.
Number of pencils Sarah puts in red cup = 2 or Two.

Number of pencils Sarah’s partner puts in green cup = 2 or Two.
Number of pencils Sarah’s partner puts in red cup = 3 or Three.

Answers of both are different.

Explanation:
Total number of pencils Sarah has = 5 or Five.
IF
Number of pencils Sarah puts in green cup = 3 or Three.
Number of pencils Sarah puts in red cup = Total number of pencils Sarah has – Number of pencils Sarah puts in green cup
= 5 – 3
= 2 or Two.

Total number of pencils Sarah has = 5 or Five.
IF
Number of pencils Sarah’s partner puts in green cup = 2 or Two.
Number of pencils Sarah’s partner puts in red cup = Total number of pencils Sarah’s partner has – Number of pencils Sarah’s partner puts in green cup
= 5 – 2
= 3 or Three.

 

 

Visual Learning Bridge

Convince Me!
With 7 penguins, could 4 be inside the cave, and 4 be outside the cave? Why or why not?
Answer:
With 7 penguins, could 4 be inside the cave, and 4 be outside the cave cannot happen because the total number of penguins count does not match, it shows more (4 + 4 = 8 not 7).

Explanation:
It cannot happen because the total number of penguins count does not match, it shows more (4 + 4 = 8 not 7).

 

 

Guided Practice
How many penguins are inside and outside? Use cubes or draw a picture. Write an equation.
Answer:
Case:1:
Number of penguins inside the cave = 4 or Four.
Number of penguins outside the cave = 3 or Three.
Case:2:
Number of penguins outside the cave = 4 or Four.
Number of penguins inside the cave = 3 or Three.

Explanation:
Case:1:
Number of penguins inside the cave = 4 or Four.
Number of penguins outside the cave = 3 or Three.
Total number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 4 + 3
= 7 or Seven.

Case:2:
Number of penguins outside the cave = 4 or Four.
Number of penguins inside the cave = 3 or Three.
Total number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 3 + 4
= 7 or Seven.

 

 

Question 1.
5 penguins in all

Answer:
Total number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 3 + 2
= 5 or Five.

Explanation:
Number of penguins inside the cave = 3 or Three.
Number of penguins outside the cave = 2 or Two.
Total number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 3 + 2
= 5 or Five.

 

 

Question 2.
8 penguins in all

Answer:
Total number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 4 + 4
= 8 or Eight.

Explanation:
Number of penguins inside the cave = 4 or Four.
Number of penguins outside the cave = 4 or Four.
Total number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 4 + 4
= 8 or Eight.

 

 

 

Independent Practice
How many bats are inside and outside? Use cubes or draw a picture. Write an equation.

Question 3.
9 bats in all

Answer:
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 5 + 4
= 9 or Nine.

Explanation:
Number of bats inside the cave = 5 or Five.
Number of bats outside the cave = 4 or Four.
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 5 + 4
= 9 or Nine.

Question 4.
8 bats in all

Answer:
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 4 + 4
= 8 or Eight.

Explanation:
Number of bats inside the cave = 4 or Four.
Number of bats outside the cave = 4 or Four.
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 4 + 4
= 8 or Eight.

 

 

Question 5.
5 bats in all

Answer:
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 3 + 2
= 5 or Five.

Explanation:
Number of bats inside the cave = 3 or Three.
Number of bats outside the cave = 2 or Two.
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 3 + 2
= 5 or Five.

 

 

Question 6.
4 bats in all

Answer:
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 2 + 2
= 4  or Four.

Explanation:
Number of bats inside the cave = 2 or Two.
Number of bats outside the cave = 2 or Two.
Total number of bats in all = Number of bats inside the cave + Number of bats outside the cave
= 2 + 2
= 4  or Four.

 

Problem Solving
Solve each problem.

Question 7.
envision® STEM
8 monkeys in all. Some live in trees. Some live on the ground. Show one way.


Answer:
Total number of monkeys in all = Number of monkeys live in trees + Number of monkeys live on the ground
= 4 + 4
= 8 or Eight.

Explanation:
Number of monkeys live in trees = 4 or Four.
Number of monkeys live on the ground =4 or Four.
Total number of monkeys in all = Number of monkeys live in trees + Number of monkeys live on the ground
= 4 + 4
= 8 or Eight.

 

 

Question 8.
Reasoning
Anna draws 2 cats. She draws 5 more cats. How many cats in all? Write an equation.


Answer:
Total number of cats Anna draws in all = 7 or Seven.

Explanation:
Number of cats Anna draws = 2 or Two.
Number of cats Anna draws more = 5 or Five.
Total number of cats Anna draws in all = Number of cats Anna draws + Number of cats Anna draws more
= 2 + 5
= 7 or Seven.

 

 

Question 9.
Higher Order Thinking
Andy has 10 balls. 1 or 2 balls are inside the toy box. How many balls are outside the toy box? Tell how you know.

Answer:
Total number of outside the toy box = 8 or Eight.

Explanation:
Number of balls Andy has = 10 or Ten.
Number of balls inside the toy box = 2 or Two.
Total number of outside the toy box = Number of balls Andy has + Number of balls inside the toy box
= 10 – 2
= 8 or Eight.

 

 

Question 10.
Assessment Practice
9 birds in all. Some birds are flying. Some birds are in a tree.

Which shows one way?
A. 4 flying, 3 in a tree
B. 5 flying, 4 in a tree
C. 1 flying, 7 in a tree
D. 8 flying, 2 in a tree
Answer:
Total number of birds in all = Number of birds flying + Number of birds in a tree
= 5 + 4
= 9 or Nine.
B. 5 flying, 4 in a tree shows one way.

Explanation:
Number of birds flying = 5 or Five.
Number of birds in a tree = 4 or Four.
Total number of birds in all = Number of birds flying + Number of birds in a tree
= 5 + 4
= 9 or Nine.

 

 

Lesson 1.4 Take from

Solve and Share

There are 6 ducks. Some fly away. How many ducks are left?

Answer:
Number of ducks left = 4 or Four.

Explanation:
Total number of ducks = 6 or Six.
Number of ducks flying away = 2 or Two.
Number of ducks left = Total number of ducks – Number of ducks flying away
= 6 – 2
= 4 or Four.

 

 

Visual Learning Bridge

Convince Me!
How do cubes help you solve the problem?
Answer:
The cubes strategy is a simple tool that helps in step-by-step calculation of addition or subtraction and to solve the problem asked.

Explanation:
The cubes strategy is a simple tool that helps in step-by-step calculation of addition or subtraction and to solve the problem asked. It helps in easy calculation  and to understand the problem and solve it.

 

 

Guided Practice
Solve. Use cubes to help. Write a subtraction equation.

Question 1.
There are 6 frogs. 2 frogs jump off.

How many frogs are left?

Answer:
Number of frogs are left = 4 or Four.

Explanation:
Total number of frogs in all = 6 or Six.
Number of frogs jump off = 2 Or Two.
Number of frogs are left = Total number of frogs in all  – Number of frogs jump off
= 6 – 2
= 4 or Four.

 

 

Question 2.
There are 7 bunnies. 1 bunny hops away.

How many bunnies are left?

Answer:
Number of bunnies are left = 6 or Six.

Explanation:
Total number of bunnies in all = 7 or Seven.
Number of bunnies hops away = 1 or One.
Number of bunnies are left = Total number of bunnies in all – Number of bunnies hops away
= 7 – 1
= 6 or Six.

 

 

Independent Practice
Solve. Use cubes or draw a picture. Write a subtraction equation.

Question 3.
There are 8 bugs. 4 bugs fly away.

How many bugs are left?

Answer:
Number of bugs are left = 4 or Four.

Explanation:
Total number of bugs in all = 8 or Eight.
Number of bugs fly away = 4 or Four.
Number of bugs are left = Total number of bugs in all – Number of bugs fly away
= 8 – 4
= 4 or Four.

 

 

Question 4.
There are 9 cats. 6 cats run away.

How many cats are left?

Answer:
Number of cats are left = 3 or Three.

Explanation:
Total number of cats in all = 9 or Nine.
Number of cats run away = 6 or Six.
Number of cats are left = Total number of cats in all – Number of cats run away
= 9 – 6
= 3 or Three.

Question 5.
Higher Order thinking
There are 7 dogs. Some run away. 3 dogs are left. How many dogs ran away.

____ dogs
Answer:
Number of dogs run away = 4 or Four.

Explanation:
Total number of dogs in all = 7 or Seven.
Number of dogs left = 3 or Three.
Number of dogs run away = Total number of dogs in all – Number of dogs left
= 7 – 3
= 4 or Four.

 

Problem Solving
Solve each problem.

Question 6.
Reasoning
Lin has 9 stamps. She gives away 4 stamps. How many stamps are left?

Answer:
Number of stamps left with Lin = 5 or Five.

Explanation:
Total number of stamps Lin has in all = 9 or Nine.
Number of stamps Lin gives away = 4 or Four.
Number of stamps left with Lin = Total number of stamps Lin has in all – Number of stamps Lin gives away
= 9 – 4
= 5 or Five.

 

Question 7.
Reasoning
Gloria has 8 flowers. She gives away 5 flowers. How many flowers are left?

Answer:
Number of flowers left with Gloria = 3 or Three.

Explanation:
Total number of flowers Gloria has in all = 8 or Eight.
Number of flowers Gloria gives away = 5 or Five.
Number of flowers left with Gloria = Total number of flowers Gloria has in all – Number of flowers Gloria gives away
= 8 – 5
= 3 or Three.

 

Question 8.
Higher Order Thinking
Find the missing number. Write a subtraction story for the equation.

Answer:
Mrs. Shas has 7 balloons. She gives away 2 balloons. How many balloons are left?
Number of balloons left with Mrs. Shas = 5 or Five.

Explanation:
Mrs. Shas has 7 balloons. She gives away 2 balloons. How many balloons are left?
Total number of balloons Mrs. Shas has in all = 7 or Seven.
Number of balloons Mrs. Shas gives away = 2 or Two.
Number of balloons left with Mrs. Shas = Total number of balloons Mrs. Shas has in all – Number of balloons Mrs. Shas gives away
= 7 – 2
= 5 or Five.

Question 9.
Assessment Practice
There are 8 bees. 6 bees fly away. How many bees are left?

A. 8 – 2 = 6 bees
B. 8 – 7 = 1 bees
C. 7 – 2 = 5 bees
D. 8 – 6 = 2 bees
Answer:
Number of bees are left = 2 or Two.
D. 8 – 6 = 2 bees

Explanation:
Total number of bees in all = 8 or Eight.
Number of bees fly away = 6 or Six.
Number of bees are left = Total number of bees in all – Number of bees fly away
= 8 – 6
= 2 or Two.

Lesson 1.5 Compare Situations

There are 5 red cars and 3 blue cars. Are there more red cars or blue cars? How many more? Show how you know.

Answer:
There are 2 red cars more than the blue cars.

Explanation:
Number of red cars = 5 or Five.
Number of blue cars = 3 or Three.
Difference = Number of red cars – Number of blue cars
= 5 – 3
= 2 or Two.

 

 

Convince Mel
Can you also add to solve the problem above? Explain.
Answer:
No, you can not add to solve the problem because you need to find the difference between the blue hats and orange hats to know how many more.

Explanation:
You need to find the difference between the blue hats and orange hats to know how many more.

Guided Practice
Use cubes to help. Write an equation. Then solve.

Question 1.
6 yellow frogs 3 green frogs

How many more yellow frogs than green frogs are there? more yellow frogs

Answer:
There are 3 yellow frogs more than the green frogs.

Explanation:
Number of yellow frogs = 6 or Six.
Number of green frogs = 3 or Three.
Difference = Number of yellow frogs – Number of green frogs
= 6 – 3
= 3 or Three.

 

Independent Practice
Use cubes or draw a picture. Write an equation. Then solve.

Question 2.
3 brown dogs I black dog

How many more brown dogs than black dogs are there?

____ more brown dogs
Answer:
There are 2 brown dogs more than the black dogs.
_2__ more brown dogs.

Explanation:
Number of brown dogs = 3 or Three.
Number of black dogs = 1 or One.
Difference = Number of brown dogs – Number of black dogs
= 3 – 1
= 2 or Two.

Question 3.
7 red beads 4 green beads

How many more red beads than green beads are there?

___ more red beads
Answer:
There are 3 red beads more than the green beads.

Explanation:
Number of red beads= 7 or Seven.
Number of green beads = 4 or Four.
Difference = Number of red beads – Number of green beads
= 7 – 4
= 3 or Three.

 

Higher Order Thinking

There are more blue birds than yellow birds. Write 2 equations to show. Then solve.

Question 4.

Answer:
Equation:1:
Difference = Number of blue birds – Number of yellow birds
= 5 – 3
= 2 or Two.

Equation:2:
Number of blue birds  = Number of birds flu away + Number of yellow birds
= 2 + 3
= 5 or Five.

Explanation:
Equation:1:
Number of blue birds = 5 or Five.
Number of yellow birds = 3 or Three.
Difference = Number of blue birds – Number of yellow birds
= 5 – 3
= 2 or Two.

Equation:2:
Number of birds flu away = 2 or Two.
Number of yellow birds = 3 or Three.
Number of blue birds  = Number of birds flu away + Number of yellow birds
= 2 + 3
= 5 or Five.

 

Problem Solving
Solve each problem. Use cubes or draw a picture.

Question 5.
Number Sense 4 fish are in a tank. 2 fish are sold. How many fish are left?

Answer:
Number of fishes left in the tank = 2 or Two.

Explanation:
Number of fishes in a tank = 4 or Four.
Number of fishes sold = 2 or Two.
Number of fishes left in the tank = Number of fishes in a tank – Number of fishes sold
= 4 – 2
= 2 or Two.

 

Question 6.
Model
Luis sees 5 green frogs. He sees I blue frog. How many more green frogs than blue frogs does Luis see?

___ more green frogs
Answer:
There are 4 green frogs seen by Luis more than the blue frogs seen by her.
_4__ more green frogs.

Explanation:
Number of green frogs Luis sees = 5 or Five.
Number of blue frogs Luis sees = 1 or One.
Difference = Number of green frogs Luis sees – Number of blue frogs Luis sees
= 5 – 1
= 4 or Four.

 

Question 7.
Higher Order Thinking
Draw some yellow flowers. Draw more red flowers than yellow flowers. How many more red flowers than yellow flowers are there?

Answer:
There are 2 red flowers more than the yellow flowers.

Explanation:
Number of red flowers = 6 or Six.
Number of yellow flowers = 4 or Four.
Difference = Number of red flowers – Number of yellow flowers
= 6 – 4
= 2 or Two.

 

Question 8.
Assessment Practice
6 gray cats
4 white cats
How many more gray cats than white cats are there?
A. 2 more gray cats
B. 4 more gray cats
C. 6 more gray cats
D. 10 more gray cats

Answer:
There are 2 gray cats more than the white cats.
A. 2 more gray cats

Explanation:
Number of gray cats = 6 or Six.
Number of white cats = 4 or Four.
Difference = Number of gray cats – Number of white cats
= 6 – 4
= 2 or Two.

 

 

 

Lesson 1.6 More Compare Situations

Amy has 7 stickers. Tim has 5 stickers. Who has fewer stickers? How many fewer? Show how you know.
Answer:
Tim has 2 stickers fewer than Amy.

Explanation:
Number of  stickers Amy has = 7 or Seven.
Number of  stickers Tim has = 5 or Five.
Difference = Number of  stickers Amy has – Number of  stickers Tim has
= 7 – 5
= 2 or Two.

 

 

 

Visual Learning Bridge

Convince Me!

How is finding how many fewer like finding how many more?
Answer:
Finding how many fewer is same like finding how many more because the process is to find how much and how many.

Explanation:
Finding how many fewer is same like finding how many more because the process is to find how much and how many. They are just in difference of calculation terms one is adding and other is subtracting.

 

Guided Practice
Use cubes to help. Write an equation. Then solve.

Question 1.
Juan has 8 red crayons. Sue has 2 blue crayons.

How many fewer crayons does Sue have than Juan?

Answer:
6 fewer crayons Sue has than Juan has.

Explanation:
Number of  red crayons Juan has = 8 or Eight.
Number of  blue crayons Sue has = 2 or Two,.
Difference = Number of red crayons Juan has – Number of blue crayons Sue has
= 8 – 2
= 6 or Six.

 

Question 2.
Ann has 4 purple grapes. Sam has 7 green grapes.

How many fewer grapes does Ann have than Sam?

Answer:
3 fewer grapes Ann have than Sam.

Explanation:
Number of purple grapes Ann has = 4 or Four.
Number of green grapes Sam has = 7 or Seven,.
Difference = Number of green grapes Sam has – Number of purple grapes Ann has
= 7 – 4
= 3 or Three.

 

Independent Practice
Use cubes or draw a picture. Write an equation. Then solve.

Question 3.
Emma buys 10 red apples. She buys 5 green apples.

How many fewer green apples than red apples does Emma buy?

Answer:
5 fewer green apples than red apples Emma buys.

Explanation:
Number of red apples Emma buys = 10 or Ten.
Number of green apples Emma buys = 5 or Five.
Difference = Number of red apples Emma buys – Number of green apples Emma buys
= 10 – 5
= 5 or Five.

 

Question 4.
Beth writes on 3 cards. Joe writes on 9 cards.

How many fewer cards does Beth write on than Joe?

Answer:
6 fewer cards Beth writes on than Joe.

Explanation:
Number of cards Beth writes = 3 or Three.
Number of cards Joe writes = 9 or Nine.
Difference = Number of cards Joe writes – Number of cards Beth writes
= 9 – 3
= 6 or Six.

 

 

Question 5.
Higher Order Thinking
There are fewer white kites than blue kites. Write 2 equations to show. Then solve.

Answer:
Equation:1:
Difference = Number of blue kites – Number of white kites
= 13 – 8
= 5 or Five.

Equation:2:
Number of white kites = Number of blue kites + Difference between the white and blue kites
= 8 + 5
= 13 or Thirteen.

Explanation:
Equation:1:
Number of white kites = 8 or Eight.
Number of blue kites = 13 or Thirteen.
Difference = Number of blue kites – Number of white kites
= 13 – 8
= 5 or Five.

Equation:2:
Number of white kites = ??
Number of blue kites = 13 or Thirteen.
Difference between the white and blue kites = 5 or Five.
Number of white kites = Number of blue kites + Difference between the white and blue kites
= 8 + 5
= 13 or Thirteen.

Problem Solving
Question 6.
Reasoning
Leah has 3 pens. Scott has 6 pens. How many pens do they have in all?

Answer:
Total number of pens they have in all = 9 or Nine.

Explanation:
Number of pens Leah has = 3 or Three.
Number of pens Scott has = 6 or Six.
Total number of pens they have in all = Number of pens Leah has + Number of pens Scott has
= 3 + 6
= 9 or Nine.

 

 

Question 7.
Reasoning
There are 7 oranges on a branch. 3 oranges fall off.

How many oranges are left?

Answer:
Number of oranges left = 4 or Four.

Explanation:
Number of oranges on a branch = 7 or Seven
Number of oranges fell off = 3 or Three.
Number of oranges left = Number of oranges on a branch – Number of oranges fell off
= 7 – 3
= 4 or Four.

 

 

Question 8.
Higher Order Thinking
Draw some blue balloons. Draw fewer yellow balloons. How many fewer yellow balloons than blue balloons are there?

Answer:
There are 2 fewer yellow balloons than blue balloons.

Explanation:
Number of blue balloons = 5 or Five.
Number of yellow balloons = 3 or Three.
Difference =  Number of blue balloons – Number of yellow balloons
= 5 – 3
= 2 or Two.

 

 

Question 9.
Assessment Practice
8 apple trees
6 pear trees
How many fewer pear trees than apple trees are there?
A. 2 fewer pear trees
B. 3 fewer pear trees
C. 6 fewer pear trees
D. 8 fewer pear trees
Answer:
There are 2 fewer pear trees than apple trees.
A. 2 fewer pear trees

Explanation:
Number of apple trees = 8 or eight.
Number of  pear trees = 6 or Six.
Difference =  Number of apple trees – Number of  pear trees
= 8 – 6
= 2 or Two.

 

Lesson 1.7 Change Unknown

Solve and Share

There are 5 train cars. More train cars join. Now there are 9 train cars. How many train cars joined?

__ trains cars joined.
Answer:
Number of trains cars joined = 4 or Four.
__4__ trains cars joined.

Explanation:
Number of train cars = 5 or Five.
Number of train cars now = 9 or nine.
Number of trains cars joined = Number of train cars now – Number of train cars
= 9 – 5
= 4 or Four.

 

Visual Learning Bridge

Convince Me!

Can you also subtract to solve the problem above? Explain.
Answer:
Yes, you can subtract to solve the problem above.
Number of cars = 5 or Five
Number of cars now = 9 or Nine.
Number of cars came more = Number of cars now – Number of cars
= 9 – 5
= 4 or Four.

Explanation:
Number of cars = 5 or Five
Number of cars now = 9 or Nine.
Difference:
Number of cars came more = Number of cars now – Number of cars
= 9 – 5
= 4 or Four.

 

Guided Practice
Use cubes to help. Write an equation. Then solve.

Question 1.
Bobby has 4 fish.

He buys more fish. Now he has 7 fish. How many fish did Bobby buy?

Answer:
Number of fishes Bobby buys = 3 or Three.

Explanation:
Number of fishes Bobby has = 4 or Four.
Number of fishes Bobby has now = 7 or seven.
Number of fishes Bobby buys = Number of fishes Bobby has now – Number of fishes Bobby has
= 7 – 4
= 3 or Three.

 

Independent Practice

Use cubes or draw a picture. Write an equation. Then solve.

Question 2.
Mary has 4 stickers.

Pat gives her more stickers. Now Mary has 8 stickers. How many stickers did Pat give Mary?

Answer:
Number of stickers Pat gives her more = 4 or Four.

Explanation:
Number of stickers Mary has = 4 or Four.
Number of stickers Mary has now = 8 or Eight.
Number of stickers Pat gives her more = Number of stickers Mary has now – Number of stickers Mary has
= 8 – 4
= 4 or Four.

 

 

Question 3.
Billy draws 4 red cars.

Then he draws some blue cars. Now there are 10 cars. How many blue cars did Billy draw?

Answer:
Number of blue cars Billy draws = 6 or Six.

Explanation:
Number of red cars Billy draws = 4 or Four.
Number of cars Billy has now = 10 or Ten.
Number of blue cars Billy draws = Number of cars Billy has now – Number of red cars Billy draws
= 10 – 4
= 6 or Six.

 

Question 4.
Higher Order Thinking
Some girls are on the bus. 2 boys get on the bus. Now there are 7 children on the bus. How many girls are on the bus? Write 2 equations to show. Then solve.

Answer:
Equation:1:
Number of girls are on the bus = Number of children on the bus now – Number of boys get on the bus
= 7 – 2
= 5 or Five.

Equation:2:
Number of children on the bus now = Number of boys get on the bus + Number of girls are on the bus
=> 7 = 2 + Number of girls are on the bus
=> 7 – 2 = Number of girls are on the bus
=> 5 or Five = Number of girls are on the bus.

Explanation:
Equation:1:
Number of boys get on the bus = 2 or Two.
Number of children on the bus now =7 or Seven.
Number of girls are on the bus = Number of children on the bus now – Number of boys get on the bus
= 7 – 2
= 5 or Five.

Equation:2:
Number of children on the bus now =7 or Seven.
Number of boys get on the bus = 2 or Two.
Number of girls are on the bus = ??
Number of children on the bus now = Number of boys get on the bus + Number of girls are on the bus
=> 7 = 2 + Number of girls are on the bus
=> 7 – 2 = Number of girls are on the bus
=> 5 = Number of girls are on the bus.

 

Problem Solving
Solve each problem.

Question 5.
6 pencils are on the desk.

Bob adds more pencils. Now there are 9 pencils. How many pencils did Bob add?
Answer:
Number of pencils Bob added more = 3 or Three.

Explanation:
Number of pencils are on the desk = 6 or Six.
Number of pencils now = 9 or Nine.
Number of pencils Bob added more = Number of pencils now – Number of pencils are on the desk
= 9 – 6
= 3 or Three.

 

Question 6.
Use Tools
Nora has 3 pretzels and 7 crackers. How many snacks does she have in all?

Answer:
Number of snacks Nora has in all = 10 or Ten.

Explanation:
Number of pretzels Nora has = 3 or Three.
Number of crackers Nora has = 7 or Seven.
Number of snacks Nora has in all = Number of pretzels Nora has + Number of crackers Nora has
= 3 + 7
= 10 or Ten.

Question 7.
Higher Order Thinking
Some yellow birds are in a tree. Some blue birds join them. Now there are 5 birds in the tree. How many yellow birds and blue birds could there be?

___ yellow birds and _____ blue birds
Answer:
Case:1: _2__ yellow birds and _3____ blue birds
Total number of birds in a tree = Number of yellow birds in a tree + Number of blue birds joined them
= 2 + 3
= 5 or Five.

Case:2: __3__ yellow birds and __2___ blue birds
Total number of birds in a tree = Number of yellow birds in a tree + Number of blue birds joined them
= 3 + 2
= 5 or Five.

Explanation:
Case:1:
Number of yellow birds in a tree = 2 or Two.
Number of blue birds joined them = 3 or Three.
Total number of birds in a tree = Number of yellow birds in a tree + Number of blue birds joined them
= 2 + 3
= 5 or Five.

Case:2:
Number of yellow birds in a tree= 3 or Three.
Number of blue birds joined them  = 2 or Two.
Total number of birds in a tree = Number of yellow birds in a tree + Number of blue birds joined them
= 3 + 2
= 5 or Five.

 

Question 8.
Assessment Practice
4 puppies play. More puppies join them. Now there are 7 puppies. How many puppies joined?
A. 5 puppies
B. 4 puppies
C. 3 puppies
D. 2 puppies
Answer:
Total number of puppies joined = 3 or Three.
C. 3 puppies

Explanation:
Number of puppies play = 4 or Four.
Number of puppies now  = 7 or Seven.
Total number of puppies joined = Number of puppies now + Number of puppies play
= 7 – 4
= 3 or Three.

 

 

Lesson 1.8 Practice Adding and Subtracting

Solve & Share

5 pebbles are brown. The other pebbles are black. There are 7 pebbles in all. How many black pebbles are there?

Answer:
Number of black pebbles = 2 or Two.

Explanation:
Number of brown pebbles = 5 or Five.
Number of pebbles in all = 7 or Seven.
Number of black pebbles =  Number of pebbles in all – Number of brown pebbles
= 7 – 5
= 2 or Two.

 

Visual Learning Bridge

Convince Me!
Compare the addition and subtraction equations. What do you notice about the numbers?

Answer:
Its noticed that both the addition equation and subtraction equation results of finding the girls strength are same.

Explanation:
Addition equation:
Number of boys in the class = 5 or Five.
Total number of students in the class = 8 or Eight.
Number of girls in the class + Number of boys in the class = Total number of students in the class
=> ?? + 5 = 8.
=> 3 + 5 = 8

Subtraction equation:
Number of boys in the class = 5 or Five.
Total number of students in the class = 8 or Eight.
Number of girls in the class = Total number of students in the class – Number of boys in the class
= 8 – 5
= 3 or Three.

 

Guided Practice
Use cubes to help.
Write an equation. Then solve.

Question 1.
Nick has 9 robots. 3 of the robots can talk.

How many robots cannot talk?

Answer:
Number of robots cannot walk = 6 or Six.

Explanation:
Number of robots Nick has = 9 or Nine.
Number of robots can walk = 3 or Three.
Number of robots Nick has = Number of robots can walk + Number of robots cannot walk
=> 9 = 3 + ??
=> 9 = 3 + 6

Question 2.
6 children play at the beach. 2 children are girls.

How many are boys?

Answer:
Number of boys playing at the beach = 4 or Four.

Explanation:
Number of girls playing at the beach = 2 or Two.
Total number of children playing at the beach = 6 or Six.
Number of boys playing at the beach = Total number of children playing at the beach – Number of girls playing at the beach
= 6 – 2
= 4 or Four.

 

 

Independent Practice
Use cubes or draw a picture. Write an equation. Then solve.

Question 3.
Jill has 9 cards. 5 cards are soccer cards.

The rest are baseball cards. How many baseball cards are there?

___ baseball cards
Answer:
Number of baseball cards = 4 or Four.
__4__ baseball cards.

Explanation:
Total number of cards Jill has = 9  or Nine.
Number of soccer cards = 5 or Five.
Number of baseball cards = Total number of cards Jill has – Number of soccer cards
= 9 – 5
= 4 or Four.

 

 

Question 4.
Rita has 7 Shells.

3 shells are big. The rest are small. How many small shells does Rita have?

__ small shells
Answer:
Number of shells are small = 4 or Four.
_4_ small shells.

Explanation:
Number of shells are big = 3 or Three.
Total number of shells Rita has = 7 or Seven.
Number of shells are small = Total number of shells Rita has – Number of shells are big
= 7 – 3
= 4 or Four.

 

 

Question 5.
Higher Order Thinking
Henry has 6 candles on his cake. I candle is green. The rest are blue. How many blue candles are there? Write 2 equations to show. Then solve.

Answer:
Equation:1:
Number of blue candles = Total number of candles on Henry cake – Number of green candles
= 6 – 1
= 5 or five.

Equation:2:
Number of green candles  + Number of blue candles = Total number of candles on Henry cake
=> 1 + ?? = 6
=> 1 + 5 = 6.

Number of blue candles = 5 or five.

Explanation:
Equation:1:
Number of green candles = 1 or One.
Total number of candles on Henry cake = 6 or Six.
Number of blue candles = Total number of candles on Henry cake – Number of green candles
= 6 – 1
= 5 or five.

Equation:2:
Number of green candles = 1 or One.
Total number of candles on Henry cake = 6 or Six.
Number of green candles  + Number of blue candles = Total number of candles on Henry cake
=> 1 + ?? = 6
=> 1 + 5 = 6.

 

 

Problem Solving
Solve each problem.

Question 6.
Make Sense
Joe buys 2 red fish. He buys some blue fish. He buys 9 fish in all. How many blue fish does Joe buy?

____ blue fish
Answer:
Number of blue fishes Joe buys = 7 or Seven.
__7__ blue fish.

Explanation:
Number of red fishes Joe buys = 2  or Two.
Total number of fishes Joe buys in all  = 9 or Nine.
Number of blue fishes Joe buys = Total number of fishes Joe buys in all – Number of red fishes Joe buys
= 9 – 2
= 7 or Seven.

 

Question 7.
Make Sense
Rachel has 8 nickels. She gives away 4 nickels. How many nickels are left?

__ nickels
Answer:
Number of nickels left = 4 or Four.
__4__ nickels.

Explanation:
Number of nickels Rachel gives away = 4 or Four.
Total number of nickels Rachel has in all  = 8 or Eight.
Number of nickels left = Total number of nickels Rachel has in all  – Number of nickels Rachel gives away
= 8 – 4
= 4 or Four.

Question 8.
Higher Order Thinking
Nina has 8 stuffed animals. Some are bears. Some are tigers. How many of each animal could Nina have?

__ bears and __ tigers
Answer:
Nina can have 4 bears and 4 tigers each.
_4_ bears and _4_ tigers.

Explanation:
Total number of stuffed animals Nina has = 8 or Eight.
Number of bears + Number of tigers = Total  number of stuffed animals Nina has
=> ?? + ?? = 8
=> 4 + 4 = 8.

Question 9.
Assessment Practice
Liz and Mary have 7 fish in all. Liz has 2 fish. How many fish does Mary have? Which equation matches the story?
A. 9 – 2 = 7 fish
B. 7 – 1 = 6 fish
C. 7 – 2 = 5 fish
D. 8 – 7 = 1 fish

Answer:
Number of fishes Mary has = 5 or Five.
C. 7 – 2 = 5 fish

Explanation:
Number of fishes Liz has = 2 or Two.
Number of fishes Liz and Mary have in all = 7 or Seven.
Number of fishes Mary has = Number of fishes Liz and Mary have in all – Number of fishes Liz has
= 7 – 2
= 5 or Five.

 

Lesson 1.9 Construct Arguments

Do you add or subtract to solve the problem? Tell why. Show how to solve. Use pictures, numbers, or words

Answer:
Number of more rabbits than turtles = 4 or Four.

Explanation:
Number of rabbits = 7 or Seven.
Number of turtles = 3 or Three.
Number of more rabbits than turtles = Number of rabbits – Number of turtles
= 7 – 3
= 4 or Four.

 

 

Thinking Habits
How can I use math to explain my work?
Is my explanation clear?

Answer:
I can use math to explain my work by taking the support of pictures ,diagrams to explain to solve the problem in a simple and easy way.
Yes, my is explanation clear.

Explanation:
I can use math to explain my work by taking the support of pictures ,diagrams to explain to solve the problem in a simple and easy way.
Yes, my is explanation clear because it is simple and easy to understand.

 

Visual Learning Bridge

Convince Me!

Look at the two ways to find the number of red crayons. How are the ways alike? How are the ways different?

Answer:
The two ways to find the number of red crayons are alike because the process is done to find the count how many are red crayons. The ways are different because the process differs as one is addition and other is subtraction.

Explanation:
Way:1:
Number of blue crayons = 6 or Six.
Total number of crayons in all = 9 or Nine.
Number of red crayons = Total number of crayons in all – Number of blue crayons
= 9 – 6
= 3 or Three.

Way:2:
Number of blue crayons = 6 or Six.
Total number of crayons in all = 9 or Nine.
Number of red crayons + Number of blue crayons = Total number of crayons in all
=> ?? + 6 = 9
=> 3 + 6 = 9.

 

 

Guided Practice
Solve. Use pictures, numbers, or words to explain.

Question 1.
Manny draws 6 tiles. 4 tiles are red. The others are green. How many green tiles does Manny draw?

Answer:
Number of green tiles Manny draws = 2 or Two.

Explanation:
Number of tiles Manny draws = 6 or Six.
Number of red tiles Manny draws = 4 or Four.
Number of green tiles Manny draws = Number of tiles Manny draws – Number of red tiles
= 6 – 4
= 2 or Two.

 

Independent Practice
Solve. Use pictures, numbers, or words to explain.

Question 2.
Jan has 8 pennies. She spends 5 pennies. How many pennies does Jan have left?
Answer:
Number of pennies left = 3 or Three.

Explanation:
Number of pennies Jan has = 8 or Eight.
Number of pennies Jan spends = 5 or Five.
Number of pennies left = Number of pennies Jan has – Number of pennies Jan spends
= 8 – 5
= 3 or Three.

 

Question 3.
Lidia has 7 pencils. Jon has 2 pencils. Who has fewer pencils? How many fewer?
Answer:
Jon has fewer pencils by 5 or five.

Explanation:
Number of  pencils Lidia has = 7 or Seven
Number of pencils Jon has = 2 or Two.
Difference =Number of  pencils Lidia has – Number of pencils Jon has
= 7 – 2
= 5 or Five.

 

 

Question 4.
Higher Order Thinking
Max has 3 apples. He buys 2 more apples. He gives away 4 apples. How many apples does Max have left? Explain.
Answer:
Number of apples left = 1 or One.

Explanation:
Number of apples Max has = 3 or Three.
Number of more apples Max buys = 2 or Two.
Total number of apples Max has = Number of apples Max has + Number of more apples Max buys
= 3 + 2
= 5 or Five.

Number of apples Max gives away = 4 or Four.
Number of apples left = Total number of apples Max has – Number of apples Max gives away
= 5 – 4
= 1 or One.

 

Problem Solving
Performance Task
Lemonade Stand
Some friends sell lemonade. Solve each problem. Use pictures, numbers, or words to explain.

Question 5.
Explain
Alex sells 3 cups. Mark sells 5 cups. How many cups do they sell in all? Here is Alex’s work.

Is his work correct? Tell why.
Answer:
Total number of cups sold in all = 8 or Eight.
His work is correct has the answer is correct.

Explanation:
Number of cups Alex sells = 3 or Three.
Number of cups Mark sells = 5 or Five.
Total number of cups sold in all = Number of cups Alex sells + Number of cups Mark sells
= 3 + 5
= 8 or Eight.

 

 

Question 6.
Be Precise
Mia sells 2 cups.
Gina sells 6 cups.
How many more cups does Gina sell than Mia?
Answer:
4 more cups Gina sells than Mia.

Explanation:
Number of cups Mia sells = 2 or Two.
Number of cups Gina sells = 6 or Six.
Difference = Number of cups Gina sells – Number of cups Mia sells
= 6 – 2
= 4 or Four.

 

Fluency Review Activity

Color these sums and differences. Leave the rest white.

Answer:

Vocabulary Review
Understand Vocabulary

Question 1.
Write an addition equation

Answer:
Addition equation:
2  +  2  =  4.

Explanation:
Addition equation:

Question 2.
Write a subtraction equation.

Answer:
Subtraction equation:
8  –  3  =  5.

Explanation:
Subtraction equation:

 

Question 3.
Circle the difference
8 – 2 = 6
Answer:

Explanation:
Difference:
8 – 2 = 6.

 

 

Question 4.
Circle one part.
5 + 3 = 8
Answer:

Explanation:
Addition Equation:
5 + 3 = 8.

 

 

Question 5.
Circle the plus sign.
3 + 4 = 7
Answer:

Explanation:
Addition Equation:
3 + 4 = 7.

 

Use Vocabulary in Writing

Question 6.
Tell how to find 8-4. Use at least one word from the Word List.
Answer:
Eight – Four = Four.

Explanation:
Eight – Four = Four.
8- 4 = 4 or Four.

 

Set A
You can solve problems about adding to.
3 turtles
1 more joins.
How many turtles now?

Answer:
Number of turtles now = 4 or Four.

Explanation:
Number of turtles = 3 or Three.
Number of turtles more joined = 1 or One.
Number of turtles now = Number of turtles + Number of turtles more joined
= 3 + 1
= 4 or Four.

 

Solve. Use cubes or draw a picture.

Question 1.
5 flowers
2 more flowers
How many flowers now?

Answer:
Number of flowers now = 7 or Seven.

Explanation:
Number of flowers = 5 or Five.
Number of more flowers = 2 or Two.
Number of flowers now = Number of flowers + Number of more flowers
= 5 + 2
= 7 or Seven.

 

 

Set B
You can solve problems about putting together.
3 red markers and 2 blue markers How many markers in all?

Answer:
Number of markers in all = 5 or Five.

Explanation:
Number of red markers = 3 or Three.
Number of blue markers = 2 or Two.
Number of markers in all = Number of red markers + Number of blue markers
= 3 + 2
= 5 or Five.

Solve. Use cubes or draw a picture.

Question 2.
4 red cars and 2 blue cars
How many cars in all?

Answer:
Number of cars in all = 6 or Six.

Explanation:
Number of red cars = 4 or Four.
Number of blue cars = 2 or Two.
Number of cars in all = Number of red cars + Number of blue cars
= 4 + 2
= 6 or Six.

 

 

Set C
You can solve problems with both addends unknown.
7 penguins in all . Some are inside a cave. Some are outside. Here is one way.

Use cubes or draw a picture. Write an equation to solve.
Answer:
Number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 5 + 2
= 7 or Seven.

Explanation:
Number of penguins inside the cave = 2 or Two.
Number of penguins outside the cave = 5 or Five.
Number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 2 + 5
= 7 or Seven.

Question 3.
6 penguins in all Some are inside. Some are outside. Show one way.

Answer:
Number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 2 + 4
= 6 or Six.

Explanation:
Number of penguins inside the cave = 2 or Two.
Number of penguins outside the cave = 4 or Four.
Number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
= 2 + 4
= 6 or Six.

 

Question 4.
9 penguins in all Some are inside. Some are outside. Show one way.

Answer:
Number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
=> 9 = 4 + 5
Number of penguins inside the cave = 4 or Four.
Number of penguins outside the cave = 5 or Five.

Explanation:
Number of penguins in all = 9 or Nine.
Number of penguins inside the cave = 4 or Four.
Number of penguins outside the cave = 5 or Five.
Number of penguins in all = Number of penguins inside the cave + Number of penguins outside the cave
=> 9 = 4 + 5

 

Set D

You can solve problems about taking from.

There are 6 pears. Mia takes 3 pears away. How many pears are left?

Answer:
Number of pears are left = 3 or Three.

Explanation:
Number of pears in all = 6 or Six.
Number of pears Mia takes away = 3 or Three.
Number of pears are left = Number of pears in all – Number of pears Mia takes away
= 6 – 3
= 3 or Three.

 

Use cubes or draw a picture. Write an equation and solve.

Question 5.
There are 7 carrots.
3 carrots are picked.
How many carrots are left?

Answer:
Number of carrots are left = 4 or Four.

Explanation:
Number of carrots in all = 7 or Seven.
Number of carrots picked away = 3 or Three.
Number of carrots are left = Number of carrots in all – Number of carrots picked away
= 7 – 3
= 4 or Four.

 

 

Set E
You can solve problems about comparing.
4 blue pens
3 yellow pens
How many more blue pens than yellow pens are there?

Answer:
There are 1 more blue pens than yellow pens.

Explanation:
Number of yellow pens = 3 or Three.
Number of blue pens = 4 or Four.
Difference = Number of blue pens – Number of yellow pens
= 4 – 3
= 1 or One.

Use cubes or draw a picture. Write an equation and solve.

Question 6.
4 black pens and I red pen
How many more black pens than red pens?

Answer:
There are 3 more black pens than red pens.

Explanation:
Number of black pens = 4 or Four.
Number of red pens = 1 or One.
Difference = Number of black pens – Number of red pens
= 4 – 1
= 3 or Three.

 

 

Question 7.
3 baseballs and 7 soccer balls
How many fewer baseballs than soccer balls?

Answer:
4 fewer baseballs than soccer balls.

Explanation:
Number of baseballs = 3 or Three.
Number of soccer balls = 7 or Seven.
Difference = Number of soccer balls – Number of baseballs
= 7 – 3
= 4 or Four.

 

 

Set F

You can find a missing addend to solve problems.

Ty has 4 grapes.
He takes some more grapes.
Now he has 9 grapes
How many grapes did Ty take?

Use cubes or draw a picture. Write an equation and solve.
Answer:
Number of grapes ty takes more = 5 or Five.

Explanation:
Number of grapes ty has = 4 or Four.
Total number of grapes ty has = 9 or Nine.
Number of grapes ty takes more = Total number of grapes ty has – Number of grapes ty has
= 9 – 4
= 5 or Five.

 

Question 8.
Ivy has 2 fish in a bowl. She adds some more fish. Now Ivy has 5 fish. How many fish did she add?

Answer:
Number of fishes Ivy has added = 3 or Three.

Explanation:
Number of fishes Ivy has in a bowl = 2 or Two.
Total number of fishes Ivy has in a bowl = 5 or Five.
Number of fishes Ivy has added = Total number of fishes Ivy has in a bowl – Number of fishes Ivy has in a bowl
= 5 – 2
= 3 or Three.

 

Set G
You can add or subtract to find a missing part.

Tom has 9 shirts.
He has 4 red shirts.
The rest are blue.
How many blue shirts does Tom have?

Use cubes or draw a picture. Write an equation and solve.
Answer:
Number of blue shirts Tom has = Total number of shirts Tom has -Number of red shirts Tom has
= 9 – 4
= 5 or Five.

Explanation:
Total number of shirts Tom has = 9 or Nine.
Number of red shirts Tom has = 4 or Four.
Number of blue shirts Tom has = Total number of shirts Tom has -Number of red shirts Tom has
= 9 – 4
= 5 or Five.

Question 9.
Gigi has 8 pairs of shoes. 4 pairs are tennis shoes. The rest are sandals. How many pairs are sandals?

Answer:
Number of pair of sandals Gigi has = 4 or Four.

Explanation:
Total number of pairs of shoes Gigi has = 8 or Eight.
Number of pairs of tennis shoes Gigi has = 4 or Four.
Number of pair of sandals Gigi has = Total number of pairs of shoes Gigi has – Number of pairs of tennis shoes Gigi has
= 8 – 4
= 4 or Four.

 

 

Set H

Thinking Habits
Construct Arguments
How can I use math to explain my work?
Is my explanation clear?

Answer:
You can use math in easy and simplify the calculations of the problem to explain your work.
Yes, your explanation is clear and easy.

Explanation:
Using math helps in making easy calculations and understand to solve the problem in an easy way.

 

 

Solve. Use pictures, numbers, or words to explain.
Question 10.
Luc has 8 fish.
He gives away 4 fish.
How many fish does Luc have left?
Answer:
Number of fishes Luc has left = 4 or Four.

Explanation:
Total number of fishes Luc has = 8 or Eight.
Number of fishes Luc gives away = 4 or Four.
Number of fishes Luc has left = Total number of fishes Luc has – Number of fishes Luc gives away
= 8 – 4
= 4 or Four.

 

 

Topic 1 Assessment Practice

Question 1.
There are 8 penguins.
Some go inside the cave.
Some stay outside.
Match the number of penguins inside the cave with the number of penguins outside.


Answer:
Number of penguins inside the cave = 4 or Four.
Number of penguins inside the cave = 5 or Five.

Explanation:
Total number of penguins  = 8 or Eight.
Total number of penguins  =  Number of penguins  inside the cave + Number of penguins outside the cave
=> 8 = ?? + ??
=> 8 = 4 + 5

 

Question 2.
Sage had 10 peppers. She cooks 3 of them. How many peppers are left?

Write a subtraction equation to solve.

Answer:
Number of peppers left = 7 or Seven.

Explanation:
Number of peppers Sage had = 10 or Ten.
Number of peppers Sage had cooked = 3 or Three.
Number of peppers left = Number of peppers Sage had – Number of peppers Sage had cooked
= 10 – 3
= 7 or Seven.

 

 

Question 3.
Sara has 5 green beads and 3 red beads. How many beads does she have in all?

Write an addition equation to solve.

Answer:
Total number of beads Sara has in all = 7 or Seven.

Explanation:
Number of beads Sara has = 10 or Ten.
Number of beads Sara has = 3 or Three.
Total number of beads Sara has in all  = Number of beads Sara has – Number of beads Sara has
= 10 – 3
= 7 or Seven.

 

 

Question 4.
Trina has 8 markers. Then she gives 5 markers to David.

Which equation shows how many markers Trina has left?
A. 7 – 2 = 5
B. 7 – 3 = 4
C. 8 – 5 = 3
D. 9 – 3 = 6
Answer:
Number of markers Trina has left = 3 or Three.
C. 8 – 5 = 3

Explanation:
Total number of markers Trina has = 8 or Eight.
Number of markers given to David = 5 or Five.
Number of markers Trina has left =Total number of markers Trina has – Number of markers given to David
= 8 – 5
= 3 or Three.

 

 

Question 5.
George had 7 postcards. Then he gets some more. Now he has 9 postcards.

Which equation does NOT describe the story?
A. 7 + 2 = 9
B. 6 + 3 = 9
C. 9 – 7 = 2
D. 9 – 2 = 7
Answer:
B. 6 + 3 = 9 does NOT describe the story.

Explanation:
Number of postcards George had = 7 or Seven.
Total number of postcards now he has = 9 or Nine.
Equation: A:
Number of postcards George had + Number of postcards George gets more  = Total number of postcards now he has
=> 7 + 2
= 9 or Nine.

Equation: C:
Number of postcards George gets more = Total number of postcards now he has – Number of postcards George had
= 9 – 7
= 2 or Two.

Equation: D:
Total number of postcards now he has – Number of postcards George gets more = Number of postcards George had
= 9 – 2
= 7 or Seven.

 

 

Question 6.
Dante has 5 books. He wants to have 7 books. How many more books does Dante need to have 7 in all?

Answer:
Number of books he needs more to have 7 in all = 2 or Two.

Explanation:
Number of books Dante has = 5 or Five.
Number of books he want to have = 7 or Seven.
Number of books he needs more to have 7 in all = Number of books he want to have – Number of books Dante has
= 7 – 5
= 2 or Two.

 

 

Question 7.
Lucy and Ellie have 6 cubes in all.
Ellie has 5 cubes.

How many cubes does Lucy have?
Choose three equations that show the story.

Answer:
Number of cubes Lucy has = 1 or one.

Explanation:
Equation:1:
Number of cubes Lucy and Ellie have in all = 6 or Six.
Number of cubes Ellie has = 5 or Five.
Number of cubes Lucy has = Number of cubes Lucy and Ellie have in all – Number of cubes Ellie has
= 6 – 5
= 1 or one.
Equation:2:
Number of cubes Lucy and Ellie have in all + Number of cubes Lucy has = Number of cubes Lucy and Ellie have in all
= 5 + 1
= 6 or Six.
Equation:3:
Number of cubes Lucy and Ellie have in all – Number of cubes Ellie has = Number of cubes Lucy has
= 6-1
= 5 or Five.

 

Question 8.
Trina has 6 ribbons. Julie has 2 ribbons. What could happen for them to have the same number of ribbons?
A. Julie gives I of her ribbons to Trina.
B. Trina gives I of her ribbons to Julie.
C. Trina gives 2 of her ribbons to Julie.
D. Trina gives 4 of her ribbons to Julie.

Answer:
Number of ribbons them to have the same number = 4 or Four.
D. Trina gives 4 of her ribbons to Julie.

Explanation:
Number of ribbons Trina has = 6 or Six.
Number of ribbons Julie has = 2 or Two.
Number of ribbons them to have the same number = Number of ribbons Trina has – Number of ribbons Julie has
= 6 – 2
= 4 or Four.

Question 9.
Draw the missing cubes on the mat.
Then write a subtraction equation that shows the story.
Owen has 5 blocks. He gives I to Jordan.
How many blocks does Owen have left?

Answer:
Number of blocks Owen have left = 4 or Four.

Explanation:
Number of blocks Owen has = 5 or Five.
Number of blocks Own gives to Jordan = 1 or One.
Number of blocks Owen have left = Number of blocks Owen has – Number of blocks Own gives to Jordan
= 5 – 1
= 4 or Four.

 

Question 10.
Hannah has 9 flowers. Carrie has 6 flowers. Which equation shows how many fewer flowers Carrie has than Hannah?


Answer:
Number of fewer flowers Carrie has than Hannah = 3 or Three.

Explanation:
Number of flowers Hannah has = 9 or Nine.
Number of flowers Carrie has = 6 or Six.
Number of fewer flowers Carrie has than Hannah = Number of flowers Hannah has – Number of flowers Carrie has
= 9 – 6
= 3 or Three.

Question 11.
Laura has 7 pears.
She wants to keep 2 pears for herself and give one to each of 6 friends.
Will Laura have enough pears?
Use pictures and words to explain.
Answer:
Laura is not having enough pears to give to her friends.
Number of pears left with her are five, which are less to give to her 6 friends each.

Explanation:
Total number of pears Laura has = 7 or Seven.
Number of pears She wants to keep for herself = 2 or Two.
Number of pears left =  Total number of pears Laura has – Number of pears She wants to keep for herself
= 7 or 2
= 5 or Five.
Number of pears She give one to each of 6 friends.
=> Number of pears required to be with her to give to her friends = 1 × 6
=> 6 or Six.
Number of pears left with her are 5, which are less to give to her friends.
Therefore, Laura does not have enough pears to give to her friends.

 

Question 12.
Nikki has 8 tennis balls. Thomas has 6 tennis balls. Which equation shows how many more tennis Nikki has than Thomas?


Answer:
Number of more tennis balls Nikki has than Thomas = 2 or Two.

Explanation:
Number of tennis balls Nikki has = 8 or Eight
Number of tennis balls Thomas has = 6 or Six.
Number of more tennis balls Nikki has than Thomas = Number of tennis balls Nikki has – Number of tennis balls Thomas has
= 8 – 6
= 2 or Two.

Topic 1 Assessment Practice

Skating Ribbons
Marta is an ice skater.
She wins ribbons for her skating.

Question 1.
Marta wins 2 blue ribbons. She wins 4 red ribbons.
How many ribbons does she win in all?
____ ribbons
Answer:
Number of ribbons Marta wins in all = 6 or Six.
_6__ ribbons.

Explanation:
Number of blue ribbons Marta wins = 2 or Two.
Number of red ribbons Marta wins = 4 or Four.
Number of ribbons Marta wins in all = Number of blue ribbons Marta wins + Number of red ribbons Marta wins
= 2 + 4
= 6 or Six.

 

Question 2.
Marta has 4 red ribbons.
She wins some more red ribbons.
Now she has 7 red ribbons.
How many more red ribbons did Marta win? ___ more red ribbons
Write an equation to show why your answer is correct.
Answer:
3 more red ribbons Marta wins.
_3__ more red ribbons.

Explanation:
Number of red ribbons Marta has = 4 or Four.
Total number of red ribbons Marta has now = 7 or Seven.
Number of red ribbons Marta wins more = Total number of red ribbons Marta has now – Number of red ribbons Marta has
= 7 – 4
= 3 or Three.

 

 

Question 3.
Marta has 8 yellow ribbons. She put some on her door. She puts the rest on her wall.

Write two different addition equations to show two ways she can put the ribbons on her door or on her wall.
Answer:
Equation:1:
Number of yellow ribbons She puts on her door = 4 or Four.
Number of yellow ribbons She puts on her wall = 4 or Four.

Equation:2:
Number of yellow ribbons She puts on her door = 5 or Five.
Number of yellow ribbons She puts on her wall = 3 or Three.

Explanation:
Equation:1:
Number of yellow ribbons Marta has = 8 or Eight.
Number of yellow ribbons Marta has = Number of yellow ribbons She puts on her door + Number of yellow ribbons She puts on her wall
= 4+ 4
= 8 or Eight.

Equation:2:
Number of yellow ribbons Marta has = 8 or Eight.
Number of yellow ribbons Marta has = Number of yellow ribbons She puts on her door + Number of yellow ribbons She puts on her wall
= 5 + 3
= 8 or Eight.

 

Question 4.
Marta has 8 yellow ribbons.
She has 2 blue ribbons.
How many more yellow ribbons than blue ribbons does Marta have?
___ more yellow ribbons
Answer:
6 more yellow ribbons than blue ribbons Marta has.
_6__ more yellow ribbons.

Explanation:
Number of yellow ribbons Marta has = 8 or Eight.
Number of blue ribbons Marta has = 2 or Two.
Difference = Number of yellow ribbons Marta has – Number of blue ribbons Marta has
= 8 – 2
= 6 or Six.

 

Question 5.
Explain why your answer to item 4 is correct. Use numbers, pictures, or words.
Answer:
My answer is correct to item 4 because:
Difference of more yellow ribbons than blue ribbons Marta has + Number of blue ribbons Marta has
=> 6 + 2
=> 8 = Number of yellow ribbons Marta has.

Explanation:
My answer is correct to item 4 because:
Difference of more yellow ribbons than blue ribbons Marta has + Number of blue ribbons Marta has
=> 6 + 2
=> 8 = Number of yellow ribbons Marta has.

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

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

TOPIC 6 GET READY!

Review What You Know!

Vocabulary

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

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

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

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

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

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

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

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

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

Statistical Measures

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Explanation:
Box plot:

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

Explanation:

Statistical Questions

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

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

Language Development

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

Answer:

PICK A PROJECT

PROJECT 6A

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

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

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

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

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

Answer:

Explanation:

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

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

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

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

Lesson 6.1 Populations and Samples

Solve & Discuss It!

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

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

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

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

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

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

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

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

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

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

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

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

Do You Understand?

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

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

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

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

Do You Know How?

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

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

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

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

Practice & Problem Solving

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

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

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

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

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

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

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

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

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

Answer:
6 marbles are expected.
Explanation:

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

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

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

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

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

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

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

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

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

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

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

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

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

Assessment Practice

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

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

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

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

Lesson 6.2 Draw Inferences from Data

Solve & Discuss It!

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

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

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

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

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

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

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

KEY CONCEPT

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

Do You Understand?

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

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

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

Do You Know How?

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

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

 

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

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

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

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

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

Practice & Problem Solving

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

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

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

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

They should order about 180 T-shirts.

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

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

They should order about 255 T-shirts

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Assessment Practice

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

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

TOPIC 6 MID-TOPIC CHECKPOINT

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

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

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

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

Answer:
45 students play exactly one sport
Explanation:

 

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

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

Topic 6 MID-TOPIC PERFORMANCE TASK

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

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

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

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

PART C

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

Lesson 6.3 Make Comparative Inferences About Populations

Explore It!

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

The table shows the results of her survey.

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

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

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

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

Try It!

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

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

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

Try It!

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

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

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

Do You Understand?

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

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

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

Do You Know How?

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

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

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

Practice & Problem Solving

Leveled Practice For 6-8, complete each statement.

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

Explanation:

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

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

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

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

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

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

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

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

Assessment Practice

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

Lesson 6.4 Make More Comparative Inferences About Populations

Explore It!

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

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

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

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

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

 

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

 

Try It!

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

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

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

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

Try It!

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

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

KEY CONCEPT

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

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

Do You Understand?

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

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

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

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

Do You Know How?

For 4 and 5, use the information below.

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

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

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

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

Practice & Problem Solving

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

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

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

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

Explanation:

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

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

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

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

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

Assessment Practice

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

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

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

3-Act Mathematical Modeling: Raising Money

ACT 1

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

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

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

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

Answer:

ACT 2

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

Answer:

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

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

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

Answer:

ACT 3

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

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

Answer:

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

Answer:

ACT 3

Extension

Reflect

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

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

SEQUEL

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

Answer:

TOPIC 6 REVIEW

Topic Essential Question

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

Vocabulary Review

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

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

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

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

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

Use Vocabulary in Writing

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

Concepts and Skills Review

LESSON 6-1 Populations and Samples

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

Practice

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

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

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

LESSON 6-2 Draw Inferences from Data

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

Practice

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

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

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

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

Practice

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

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

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

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

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

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

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

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

 

TOPIC 6 Fluency Practice

Riddle Rearranging

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: RIVER BANK

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

Go through the enVision Math Common Core Grade 8 Answer Key Topic 5 Analyze And Solve Systems Of Linear Equations regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 5 Analyze And Solve Systems Of Linear Equations

Topic Essential Question
What does it mean to solve a system of linear equations?
Answer:
A system of linear equations is just a set of two or more linear equations. In two variables (x and y), the graph of a system of two equations is a pair of lines in the plane

Topic 5 ënVision STEM Project

Did You Know?
After the Boston Tea Party of 1773, many Americans switched to drinking coffee rather than tea because drinking tea was considered unpatriotic.

Although Brazil is the largest coffee-producing nation in the world, Americans combine to drink 0.2% more coffee each year than Brazilians. The third-ranked nation for total coffee consumption, Germany, consumes approximately 44% as much coffee as either the United States or Brazil.

The United States consumes the most coffee by total weight, but Americans do not drink the most coffee per capita. People in northern European countries like Finland, Norway, and Holland drink more than twice as much coffee as their American counterparts each day.

In some coffee-producing nations, millions of acres of forest are cleared to make space for coffee farming. Sustainable farms grow coffee plants in natural growing conditions without chemicals and with minimal waste.

Coffee beans are actually seeds that are harvested from cherries that grow on coffee plants in tropical climates.

Your Task: Daily Grind

Coffee roasters create coffee blends by mixing specialty coffees with less expensive coffees in order to create unique coffees, reduce costs, and provide customers with consistent flavor. You and your classmates will explore coffee blends while considering the environmental and economic impact of the coffee trade.

Topic 6 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.
linear equation
parallel
slope
y-intercept

Question 1.
The value of m in the equation y = mx + b represents the __________ .
Answer:
We know that,
The value of m in the equation
y = mx + b
represents the “Slope”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Slope”

Question 2.
When lines are the same distance apart over their entire lengths, they are _________.
Answer:
We know that,
When lines are the same distance apart over their entire lengths, they are “Parallel”
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Parallel”

Question 3.
The _________ is the value b in the equation y = mx + b.
Answer:
We know that,
The “y-intercept” is the value of b in the equation
y = mx + b
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “y-intercept”

Question 4.
A __________ is a relationship between two variables that gives a straight line when graphed.
Answer:
We know that,
A “Linear relationship” is a relationship between two variables that gives a straight line when graphed
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is “Linear relationship”

Identifying Slope and y-Intercept

Identify the slope and the y-intercept of the equation.
Question 5.
y = 2x – 3
slope = _________
y-intercept = ________
Answer:
The given equation is:
y = 2x – 3
Compare the above equation with
y = mx + b
Where,
m is the slope
b is the y-intercept
Hence, from the above,
We can conclude that
The slope is: 2
The y-intercept is: -3

Question 6.
y =-0.5x + 2.5
slope = _________
y-intercept = ________
Answer:
The given equation is:
y = -0.5x + 2.5
Compare the above equation with
y = mx + b
Where,
m is the slope
b is the y-intercept
Hence, from the above,
We can conclude that
The slope is: -0.5
The y-intercept is: 2.5

Question 7.
y – 1 = -x
slope = _________
y-intercept = ________
Answer:
The given equation is:
y – 1 = -x
Add 1 on both sides
So,
y = -x + 1
Now,
Compare the above equation with
y = mx + b
Where,
m is the slope
b is the y-intercept
Hence, from the above,
We can conclude that
The slope is: -1
The y-intercept is: 1

Graphing Linear Equations

Graph the equation.
Question 8.
y = \(\frac{2}{3}\)x – 2

Answer:
The given equation is:
y = \(\frac{2}{3}\)x – 2
Hence,
The representation of the given equation in the coordinate plane is:

Question 9.
y = -2x + 1

Answer:
The given equation is:
y = -2x + 1
Hence,
The representation of the given equation in the coordinate plane is:

Solving Equations for Variables

Solve the equation for y.
Question 10.
y – x = 5
Answer:
The given equation is:
y – x = 5
Add x on both sides
So,
y – x + x = 5 + x
y = x + 5
Hence, from the above,
We can conclude that the value of y is:
y = x + 5

Question 11.
y + 0.2x = -4
Answer:
The given equation is:
y + 0.2x = -4
Subtract with 0.2x on both sides
So,
y + 0.2x – 0.2x = -4 – 0.2x
y = -0.2x – 4
Hence, from the above,
We can conclude that the value of y is:
y = -0.2x – 4

Question 12.
–\(\frac{2}{3}\)x + y = 8
Answer:
The given equation is:
–\(\frac{2}{3}\)x + y = 8
Add with \(\frac{2}{3}\)x on both sides
So,
–\(\frac{2}{3}\)x + \(\frac{2}{3}\)x + y = 8 + \(\frac{2}{3}\)x
y = \(\frac{2}{3}\)x + 8
Hence, from the above,
We can conclude that the value of y is:
y = \(\frac{2}{3}\)x + 8

Language Development

Complete the fishbone map by writing key terms or phrases related to systems of linear equations on each diagonal. Connect supporting ideas on the horizontal lines.

Topic 5 PICK A PROJECT

PROJECT 5A
What is the most interesting discussion you’ve had with your classmates?
PROJECT: WRITE A SPEECH FOR A DEBATE

PROJECT 5B
What can you compare with a Venn diagram?
PROJECT: DRAW A VENN DIAGRAM

PROJECT 5C
What can you do with a smartphone?
PROJECT: CHOOSE A CELL PHONE PLAN

PROJECT 5D
If you made a stained-glass window, what colors and shapes would you use?
PROJECT: MAKE A MODEL OF A STAINED-GLASS WINDOW

Lesson 5.1 Estimate Solutions by Inspection

Solve & Discuss It!
Draw three pairs of lines, each showing a different way that two lines can intersect or not intersect. How are these pairs of lines related?

I can… find the number of solutions of a system of equations by inspecting the equations.
Answer:
The representation of three pairs of lines, each showing a different way that two lines can intersect or not intersect is:

Now,
Parallel lines:
The lines that do not intersect each othse and have the same slope but different y-intercepts
Perpendicular lines:
The lines that intersect each other with the product of the slopes of lines -1 and different y-intercepts
Intersecting lines:
The lines that are neither parallel nor perpendicular are called “Intersecting lines”
Note:
All perpendicular lines should be intersecting lines but all intersecting lines should not be perpendicular lines

Focus on math practices
Look for Relationships Is it possible for any of the pairs of lines drawn to have exactly two points in common? Explain.
Answer:
We know that,
Any pair of lines should intersect at only 1 common point
Hence, from the above,
We can conclude that It is not possible for any of the pairs of lines drawn to have exactly two points in common

Essential Question
How are slopes and y-intercepts related to the number of solutions of a system of linear equations?
Answer:
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions

Try It!

How many solutions does this system of equations have? Explain.
y = x + 1
y = 2x + 2
The system of equations has ________ solution. The equations have _______ slopes, so the lines intersect at ________ point.
Answer:
The given system of equations are:
y = x + 1 —–(1)
y = 2x + 2 —-(2)
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 1, b = 1
For the second equation,
m = 2 and b = 2
Now,
When we observe the slopes and y-intercepts, they are different
We know that,
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
Hence, from the above,
We can conclude that
The given system of equations have only 1 solution
The equation has 2 slopes
The lines intersect at (-1, 0)

Convince Me!
The equations of a system have the same slopes. What can you determine about the solution of the system of equations?
Answer:
It is given that the equations of a system have the same slopes
Hence,
When slopes are the same and the y-intercepts are different, the lines may be either parallel
So,
There are no solutions
If the lines have the same slopes and the same y-intercepts, they intersect at all points,
So,
There are infinitely many solutions

Try It!

How many solutions does each system of equations have? Explain.
a. y = -3x + 5
y = -3x – 5
Answer:
The given system of equations are:
y = -3x + 5 ——(1)
y = -3x – 5 ——-(2)
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = -3, b = 5
For the second equation,
m = -3, b = -5
Now,
From the above,
We can observe that there are the same slopes but different y-intercepts
We know that,
When slopes are the same and the y-intercepts are different, the lines may be either parallel and there are no solutions
Hence, from the above,
We can conclude that there are no solutions for the given system of equations

b. y = 3x + 4
5y – 15x – 20 = 0
Answer:
The given system of equations are:
y = 3x + 4
5y – 15x – 20 = 0
Now,
5y = 15x + 20
Divide by 5 into both sides
So,
y = 3x + 4
So,
The given system of equations are:
y = 3x + 4 ——-(1)
y = 3x + 4 ——–(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 3, b = 4
For the second equation,
m = 3, b = 4
Now,
From the above,
We can observe that there are the same slopes and the same y-intercepts
We know that,
When slopes and the y-intercepts are the same,
The lines have infinitely many solutions
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

KEY CONCEPT

You can inspect the slopes and y-intercepts of the equations in a system of linear equations in order to determine the number of solutions of the system.
One Solution
y = 2x + 4
y = 3x – 1
The slopes are different. The lines intersect at 1 point.

No Solution
y = 3x + 4
y = 3x + 5
The slopes are the same, and the y-intercepts are different. The lines are parallel.

Infinitely Many Solutions
y = 3x + 4
y = 4 + 3x
The slopes are the same, and the y-intercepts are the same. The lines are the same.

Do You Understand?
Question 1.
Essential Question How are slopes and y-intercepts related to the number of solutions of a system of linear equations?
Answer:
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions

Question 2.
Construct Arguments Macy says that any time the equations in a system have the same y-intercept, the system has infinitely many solutions. Is Macy correct? Explain.
Answer:
It is given that
Macy says that any time the equations in a system have the same y-intercept, the system has infinitely many solutions.
Now,
Consider two equations with the same y-intercept and different slopes.
y = 2x + 3
y = 5x + 3
Compare the above system of equation with
y = mx + b
Hence,
We can say that
This system has only one solution.
Now,
Consider another system of equations with the same y-intercepts and the same slopes
y = x + 7
y = x + 7
Compare the above system of equation with
y = mx + b
Hence,
We can say that
This system has infinitely many solutions.
Hence, from the above,
We can conclude that Macy is not correct

Question 3.
Use Structure How can you determine the number of solutions of a system of linear equations by inspecting its equations?
Answer:
By inspecting the equations, the number of solutions can be determined as mentioned below:
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions

Do You Know How?
Question 4.
Kyle has x 3-ounce blue marbles and a 5-ounce green marble. Lara has x 5-ounce green marbles and a 3-ounce blue marble. Is it possible for Kyle and Lara to have the same number of green marbles and the same total bag weight, y? Explain.

Answer:
It is given that
Kyle has x 3-ounce blue marbles and a 5-ounce green marble. Lara has x 5-ounce green marbles and a 3-ounce blue marble.
Now,
The total number of marbles = The total number of green marbles + The total number of blue marbles
So,
For Kyle,
The total number of marbles = x+ 1
For Lara,
The total number of marbles = x + 1
Now,
From the total number of marbles,
We can observe that Kyle and Lara have the same number of marbles
Since Kyla and Lara have the same number of marbles,
The total weight of the marbles will also be the same
Hence, from the above,
We can conclude that
Kyle and Lara have the same number of green marbles and the same total bag weight, y

Question 5.
How many solutions does this system of linear equations have? Explain.
\(\frac{1}{2}\)x = y
y = \(\frac{1}{2}\)x + 3
Answer:
The given system of equations are:
\(\frac{1}{2}\)x = y ——(1)
y = \(\frac{1}{2}\)x + 3 —–(2)
Now,
Comapre the given system of equations with
y = mx + b
So,
For the first equation,
m = \(\frac{1}{2}\), b = 0
For the second equation,
m = \(\frac{1}{2}\), b = 3
We know that,
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
Hence, from the above,
We can conclude that the given system of equations have no solutions

Question 6.
How many solutions does this system of linear equations have? Explain.
3y + 6x = 12
8x + 4y = 16
Answer:
The given system of equations are:
3y + 6x = 12
8x + 4y = 16
So,
3y = -6x + 12
Divide by 3 into both sides
So,
y = -2x + 4
So,
4y = -8x + 16
Divide by 4 into both sides
So,
y = -2x + 4
So,
The required system of equations are:
y = -2x + 4 —–(1)
y = -2x + 4 —–(2)
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that the given system of equations have infinitely many solutions

Practice & Problem Solving

Question 7.
Leveled Practice Two rovers are exploring a planet. The system of equations below shows each rover’s elevation, y, at time x. What conclusion can you reach about the system of equations?

Rover A: y = 1.9x – 8
Rover B: 7y = 13.3x – 56
The slope for the Rover A equation is _________ the slope for the Rover B equation.
The y-intercepts of the equations are ___________.
The system of equations has __________ solution(s).
Answer:
It is given that
Two rovers are exploring a planet. The system of equations below shows each rover’s elevation, y, at time x.
Now,
The given system of equation are:
Rover A: y = 1.9x – 8 ——(1)
Rover B: 7y = 13.3x – 56
Now,
Divide the equation of Rover B with 7
So,
Rover B: y = 1.9x – 8 ——(2)
Now,
When we compare the given system of equations with
y = mx + b
For Rover A,
m = 1.9, b = -8
For Rover B,
m = 1.9, b = -8
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that the given system of equations ahve infinitely many solutions

Question 8.
How many solutions does this system have?
y = x – 3
4x – 10y = 6
Answer:
The given system of equations are:
y = x – 3 ——(1)
4x – 10y = 6
So,
10y = 4x – 6
Divide by 10 into both sides
So,
y = 0.4x – 0.6 ——(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 1, b = -3
For the second equation,
m = 0.4, b = -0.6
We know that,
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
Hence, from the above,
We can conclude that the given system of equations have only 1 solution

Question 9.
How many solutions does this system have?
x + 3y = 0
12y = -4x
Answer:
The given system of equations are:
x + 3y = 0
12y = -4x
Now,
3y = -x + 0
y = –\(\frac{1}{3}\)x + 0 ——(1)
12y = -4x
y = –\(\frac{4}{12}\)x
y = –\(\frac{1}{3}\)x ——-(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = –\(\frac{1}{3}\), b = 0
For the second equation,
m = –\(\frac{1}{3}\), b = 0
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that the given system of equations ahve infinitely many solutions

Question 10.
What can you determine about the solution(s) of this system?
-64x + 96y = 176
56x – 84y = -147
Answer:
The given system of equations are:
-64x + 96y = 176
56x – 84y = -147
So,
96y = 64x + 176
84y = 56x + 147
Now,
Divide by 96 into both sides
y = \(\frac{2}{3}\)x + \(\frac{11}{6}\) ——(1)
Now,
Divide by 84 into both sides
y = \(\frac{2}{3}\)x + \(\frac{7}{4}\) ——-(2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = \(\frac{2}{3}\), b = \(\frac{11}{6}\)
For the second equation,
m = \(\frac{2}{3}\), b = \(\frac{7}{4}\)
We know that,
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
Hence, from the above,
We can conclude that the given system of equations have no solutions

Question 11.
Determine whether this system of equations has one solution, no solution, or infinitely many solutions.
y = 8x + 2
y = -8x + 2
Answer:
The given system of equations are:
y = 8x + 2 —–(1)
y = -8x + 2 —–(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = 8, b = 2
For the second equation,
m = -8, b = 2
We know that,
When slopes and y-intercepts are different or y-intercepts are the same, the two lines intersect at one point,
There is only 1 solution.
Hence, from the above,
We can conclude that the given system of equations have only 1 solution

Question 12.
Construct Arguments Maia says that the two lines in this system of linear equations are parallel. Is she correct? Explain.
2x + y = 14
2y + 4x = 14
Answer:
It is given that
Maia said the below system of linear equations to be paralle
Now,
The given system of equations are:
2x + y = 14
2y + 4x = 14
Now,
y = -2x + 14 —-(1)
2y = -4x + 14
Divide by 2 into both sides
So,
y = -2x + 7 —–(2)
Now,
Compare the given system of equations with
y = mx + b
So,
For the first equation,
m = -2, b = 14
For the second equation,
m = -2, b = 7
We know that,
For the lines to be parallel, the slopes have to be the same and the y-intercepts be different
Hence, from the above,
We can conclude that Maia is correct

Question 13.
Reasoning Describe a situation that can be represented by using this system of equations. Inspect the system to determine the number of solutions and interpret the solution within the context of your situation.
y = 2x + 10
y = x + 15
Answer:
The given system of equations are:
y = 2x + 10 — (1)
y = x + 15 —–(2)
Now,
Compare the above system of equations with
y = mx + b
So,
For the first equation,
m = 2, b = 10
For the second equation,
m = 1, b = 15
We know that,
When the slopes and y-intercepts are different for a system of linear equations,
There is only 1 solution
Hence, from the above,
We can conclude that the given system of equations has only 1 solution

Question 14.
Look for Relationships Does this system have one solution, no solutions, or infinitely many solutions? Write another system of equations with the same number of solutions that uses the first equation only.
12x + 51y = 156
-8x – 34y = -104
Answer:
The given system of equations are:
12x + 51y = 156
-8x – 34y = -104
So,
51y = -12x + 156
Divide by 51 into both sides
So,
y = –\(\frac{4}{17}\)x + \(\frac{52}{17}\) —-(1)
So,
8x + 34y = 104
34y = -8x + 104
Divide by 34 into both sides
So,
y = –\(\frac{4}{17}\)x + \(\frac{52}{17}\) ——-(2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = –\(\frac{4}{17}\), b = \(\frac{52}{17}\)
For the second equation,
m = –\(\frac{4}{17}\), b = \(\frac{52}{17}\)
We know that,
When we have the same slopes and the same y-intercepts, the solutions for a system of linear equations are infinite
Now,
Another system of equations with the first equation as one of the equations is:
12x + 51y = 156
48x + 204y = 624
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

Question 15.
The equations represent the heights, y, of the flowers, in inches, after x days. What does the y-intercept of each equation represent? Will the flowers ever be the same height? Explain.

Answer:
It is given that
The equations represent the heights, y, of the flowers, in inches, after x days.
Now,
The given system of equations are:
y = 0.7x + 2 —–(1)
y = 0.4x + 2 —–(2)
Now,
Compare the above system of equations with
y = mx + b
So,
For the first equation,
m = 0.7, b = 2
For the second equation,
m = 0.4, b = 2
Now,
We know that
b represents the y-intercept or the initial value
Now,
When we compare the slopes of the equations,
We know that,
The slopes are different
Hence, from the above,
We can conclude that
The y-intercept of each equation represents the initial height of the flowers
The heights of the flowers are not the same

Question 16.
Does this system have one solution, no solution, or an infinite number of solutions?
4x + 3y = 8
8x + y = 2
Answer:
The given system of equations are:
4x + 3y = 8
8x + y = 2
Now,
3y = -4x + 8
Divide by 3 into both sides
So,
y = –\(\frac{4}{3}\)x + \(\frac{8}{3}\) —-(1)
So,
y = -8x + 2 —–(2)
Now,
Compare the above equations with
y = mx + b
So,
For the equation 1,
m = –\(\frac{4}{3}\), b = \(\frac{8}{3}\)
For the equation 2,
m = -8, b = 2
We know that,
When the slopes and y-intercepts of a system of equations are different, the system has only a solution
Hence, from the above,
We can conclude that the given system of equations has only 1 solution

Question 17.
Higher-Order Thinking Under what circumstances does the system of equations Qx + Ry = S and y = Tx + S have infinitely many solutions?
Answer:
The given system of equations are:
Qx + Ry = S
y = Tx + S
So,
Ry = -Qx + S
Divide by R into both sides
So,
y = –\(\frac{Q}{R}\)x + \(\frac{S}{R}\) —–(1)
y = Tx + S —–(2)
Now,
For the given system of equations to have infinitely many solutions,
The slopes must be the same
The y-intercepts must be the same
So,
–\(\frac{Q}{R}\) = T
\(\frac{S}{R}\) = S
Hence, from the above,
We can conclude that the given system of equations will have infinitely many solutions when
–\(\frac{Q}{R}\) = T
\(\frac{S}{R}\) = S

Assessment Practice
Question 18.
By inspecting the equations, what can you determine about the solution(s) of this system?
12y = 9x + 33
20y = 15x + 55
Answer:
The given system of equations are:
12y = 9x + 33 —-(1)
20y = 15x + 55 —-(2)
So,
Divide eq (1) by 12 into both sides
So,
y = \(\frac{3}{4}\)x + \(\frac{11}{4}\) —-(3)
So,
Divide eq (2) by 20 into both sides
So,
y = \(\frac{3}{4}\)x + \(\frac{11}{4}\) —–(4)
Now,
Compare the above equations with
y = mx + b
So,
For the 3rd equation,
m = \(\frac{3}{4}\), b = \(\frac{11}{4}\)
For the 4th equation,
m = \(\frac{3}{4}\), b = \(\frac{11}{4}\)
We know that,
When the slopes and the y-intercepts of a system of equations are different, the system has only 1 solution
Hence, from the above,
We can conclude that the given system of equations has infinitely many solutions

Question 19.
Choose the statement that correctly describes how many solutions there are for this system of equations.
y = \(\frac{2}{3}\)x + 3
y = \(\frac{5}{4}\)x + 3
A. Infinitely many solutions because the slopes are equal and the y-intercepts are equal
B. Exactly one solution because the slopes are equal but the y-intercepts are NOT equal
C. No solution because the slopes are equal and the y-intercepts are NOT equal
D. Exactly one solution because the slopes are NOT equal
Answer:
The given system of equations are:
y = \(\frac{2}{3}\)x + 3 —-(1)
y = \(\frac{5}{4}\)x + 3 —–(2)
Now,
Compare the abovee quations with
y = mx + b
So,
For the first equation,
m = \(\frac{2}{3}\), b = 3
Fpr the second equation,
m = \(\frac{5}{4}\), b = 3
We know that
When the slopes are different and the y-intercepts are the same or different, the system of equations has only 1 solution
Hence, from the above,
We can conclude that option D matches with the given system of equations

Lesson 5.2 Solve Systems by Graphing

Explore It!
Beth and Dante pass by the library as they walk home using separate straight paths.

I can… find the solution to a system of equations using graphs.

A. Model with Math The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library.

Answer:
It is given that
Beth and Dante pass by the library as they walk home using separate straight paths.
Now,
In the graph,
the location of the graph is given
Hence,
The representation of one of the paths of Beth and Dante that are passing through the library in a straight line is:

B. Write a system of equations that represents the paths taken by Beth and Dante.
Answer:
From part (a),
The representation of one of the paths of Beth and Dante that are passing through the library in a straight line is:

So,
From the graph,
The points that are passing through the path of Beth is: (-4, 0), (0, 2)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Slope (m) = \(\frac{2 – 0}{0 + 4}\)
= \(\frac{2}{4}\)
= \(\frac{1}{2}\)
We know that,
The form of the linear equation in the slope-intercept form is:
y = mx + b
So,
y = \(\frac{1}{2}\)x + b
2y = x + 2b
Substitute (-4, 0) (or) (0, 2) in the above equation
So,
2b = 4
b = 2
So,
The required equation is:
2y = x + 4
Now,
The points that are passing through the path of Dante is: (-1, -2), (0, 0)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Slope (m) = \(\frac{0 + 2}{0 + 1}\)
= \(\frac{2}{1}\)
= 2
We know that,
The form of the linear equation in the slope-intercept form is:
y = mx + b
So,
y = 2x + b
Substitute (-1, -2) (or) (0, 0) in the above equation
So,
b = 0
So,
The required equation is:
y = 2x
Hence, from the above,
We can conclude that
The required system of linear equations are:
2y = x + 4
y = 2x

Focus on math practices
Reasoning What does the point of intersection of the lines represent in the situation?
Answer:
Point of intersection means the point at which two lines intersect. These two lines are represented by the equation
a₁x + b₁y + c₁= 0 and
a₂x + b₂y + c₂ = 0, respectively.
By solving the two equations, we can find the solution for the point of intersection of two lines.

Essential Question
How does the graph of a system of linear equations represent its solution?
Answer:
Each shows two lines that make up a system of equations. If the graphs of the equations intersect, then there is one solution that is true for both equations. If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

Try It!

Solve the system by graphing.

y = 3x + 5
y = 2x + 4
The solution is the point of intersection (______, _______)
Answer:
The given system of equations are:
y = 3x + 5
y = 2x + 4
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution or the intersection point of the given system of equations is: (-1, 2)

Convince Me!
How does the point of intersection of the graphs represent the solution of a system of linear equations?
Answer:
When you graph an equation, each point (x,y) on the line satisfies the equation. Therefore, when 2 lines intersect, the coordinates of the intersection point satisfy both equations, i.e. the intersection point represents the solution of the set

Try It!

Solve each system by graphing. Describe the solutions.

a. 5x + y = -3
10x + 2y = -6
Answer:
The given system of equations are:
5x + y = -3
10x + 2y = -6
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the given system of equations has infinitely many solutions since both equations are on the same line

b. x + y = 7
2x + 6y = 12
Answer:
The given system of equations are:
x + y = 7
2x + 6y = 12
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the given system of equations has only one solution i.e., (7.5, -0.5)

KEY CONCEPT

The solution of a system of linear equations is the point of intersection of the lines defined by the equations.

Do You Understand?
Question 1.
Essential Question How does the graph of a system of linear equations represent its solution?
Answer:
Each shows two lines that make up a system of equations. If the graphs of the equations intersect, then there is one solution that is true for both equations. If the graphs of the equations do not intersect (for example, if they are parallel), then there are no solutions that are true for both equations.

Question 2.
Reasoning If a system has no solution, what do you know about the lines being graphed?
Answer:
If a system has no solution, it is said to be inconsistent. The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.

Question 3.
Construct Arguments in a system of linear equations, the lines described by each equation have the same slopes. What are the possible solutions to the system? Explain.
Answer:
If the given system of equations has the same slope, then
For the same slope and the same y-intercept – There ate infinitely many solutions
For the same slope and different outputs – There are no solutions

Do You Know How?
In 4-6, graph each system of equations and find the solution.
Question 4.
y = -3x – 5
y = 9x + 7

Answer:
The given system of equations are:
y = -3x – 5
y = 9x + 7
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (-1, -2)

Question 5.
y = 2x – 5
6x + 3y = -15

Answer:
The given system of equations are:
y = 2x – 5
6x + 3y = -15
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (0, -5)

Question 6.
y = -4x + 3
8x + 2y = 8

Answer:
The given system of equations are:
y = -4x + 3
8x + 2y = 8
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that there are no solutions for the given system of equations

Practice & Problem Solving

In 7 and 8, graph each system of equations to determine the solution.
Question 7.
x + 4y = 8
3x + 4y = 0

Answer:
The given system of equations are:
x + 4y = 8
3x + 4y = 0
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (-4, 3)

Question 8.
2x – 3y = 6
4x – y = 12

Answer:
The given system of equations are:
2x – 3y = 6
4x – y = 12
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (3, 0)

Question 9.
The total cost, c, of renting a canoe for n hours can be represented by a system of equations.

a. Write the system of equations that could be used to find the total cost, c, of renting a canoe for n hours.
Answer:
It is given that
The total cost, c, of renting a canoe for n hours can be represented by a system of equations.
Now,
The total cost for renting a canoe (y) = The cost of renting a canoe per hour × The number of hours (n) + Deposit
So,
For River Y,
y = 33n
For River Z,
y = 5n + 13
Hence, from the above,
We can conclude that the system of equations for the total cost of a canoe is:
y = 33n
y = 5n + 13

b. Graph the system of equations.

Answer:
From part (a),
The system of equations is:
y = 33n
y = 5n + 13
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution of the given system of equations is: (0.5, 16)

c. When would the total cost for renting a canoe be the same on both rivers? Explain.
Answer:
From the graph that is in part (b),
We can observe that the y-axis passes through 12
Hence,from the above,
We can conclude that the total cost for renting a canoe can be the same on both rivers after 12 hours

Question 10.
Graph the system of equations and determine the solution.
x + 2y = 4
4x + 8y = 64

Answer:
The given system of equations are:
x + 2y = 4
4x + 8y = 64
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that there are no solutions for the given system of equations since they are parallel

Question 11.
Graph the system of equations, then estimate the solution.
y = 1.5x + 1
y = -1.5x + 5.5

Answer:
The given system of equations are:
y = 1.5x + 1
y = -1.5x + 5.5
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the estimated solution for the given system of equations is: (1, 3)

In 12 and 13, graph and determine the solution of the system of equations.
Question 12.
-3y = -9x + 3
-6y = -18x – 12

Answer:
The given system of equations are:
-3y = -9x + 3
-6y = -18x – 12
So,
The representation of the given syetem of equations in the coordinate plane is:

Hence, from the above,
We can conclude that there are no solutions for the given system of equations since they are parallel lines

Question 13.
x + 5y = 0
25y = -5x

Answer:
The given system of equations are:
x + 5y = 0
25y = -5x
So,
The representation of the given system of equations i the coordinate plane is:

Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations since both equations are in the same line

Question 14.
Higher Order Thinking The total cost, c, of making n copies can be represented by a system of equations.

a. Estimate how many copies you need to make for the total cost to be the same at both stores.
Answer:
It is given that
The total cost, c, of making n copies can be represented by a system of equations.
We know that,
1 dollar = 0.1 cent
So,
The total cost of making n copies (y) = The number of copies per hour × The number of copies + the cost of machine use
So,
For Store W,
y = 5n —— (1)
For Store Z,
y = 0.20n + 2 —- (2)
Now,
From the given graph,
We can observe that the given system of equations pass through (6, 5)
Hence, from the above,
We can conclude that we have to make 6 copies for the total cost to be the same at both stores

b. If you have to make a small number of copies, which store should you go to? Explain.
Answer:
To make a small number of copies,
We will see which store gives us the less cost for printing the number of copies
So,
From the given information,
We can observe that Store Z gives us the less cost for printing the more number of copies
Hence, from the above,
We can conclude that we will goto Store Z if we have to make small number of copies

Assessment Practice
Question 15.
Consider the following system of equations.
y =-3x + 6
y = 3x – 12
Which statement is true about the system?
A. The graph of the system is a pair of lines that do not intersect.
B. The graph of the system is a pair of lines that intersect at exactly one point.
C. The graph of the system is a pair of lines that intersect at every point.
D. The system has infinitely many solutions.
Answer:
The given system of equations are:
y = -3x + 6
y = 3x – 12
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that option B matches with the description of the above graph

Question 16.
What is the solution of the system of equations?

Answer:
The given graph is:

So,
From the given graph,
We can observe that
The intersection point is: (2.5, -3)
Hence, from the above,
We can conclude that the solution for the given system of equations are: (2.5, 3)

Topic 5 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How can you determine the number of solutions of a system by looking at the equations? Lesson 5-1
Answer:
A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system has no solution. If the slopes are different, the system has one solution. If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

Question 2.
How many solutions does the system of equations have? Explain. Lesson 5-1
2x – 9y = -5
4x – y = 2
Answer:
The given system of equations are:
2x – 9y = -5
4x – y = 2
So,
9y = 2x + 5
y = \(\frac{2}{9}\)x + \(\frac{5}{9}\) —— (1)
So,
y = 4x – 2 —— (2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = \(\frac{2}{9}\), b = \(\frac{5}{9}\)
For the second equation,
m = 4, b = -2
We know that,
When the slopes and y-intercepts of a given system of equations are different, the system of equations has only one solution
Hence, from the above,
We can conclude that the given system of equations has only one solution

Question 3.
Graph the system of equations and find the solution. Lesson 5-2
y = 2x – 1
y = \(\frac{1}{2}\)x + 2

Answer:
The given system of equations are:
y = 2x – 1
y = \(\frac{1}{2}\)x + 2
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution of the given system of equations is: (2, 3)

Question 4.
One equation in a system is y =-3x + 7. Which equation gives the system no solution? Lesson 5-1
A. y = -3x + 7
B. y = 3x + 5
C. y = -3x + 5
D. y= \(\frac{1}{3}\)x – 7
Answer:
It is given that
One of the equations in a system of equations is:
y = -3x + 7
Now,
For a system of equations to have no solution,
The slopes must be equal but the y-intercepts must be different
So,
With the above description, we have 2 options matched but option A has the same slope and y-intercept as the first equation
Hence, from the above,
We can conclude that option C must be the other equation in the given system of equations

Question 5.
Finn bought 12 movie tickets. Student tickets cost $4, and adult tickets cost $8. Finn spent a total of $60. Write and graph a system of equations to find the number of student and adult tickets Finn bought. Lesson 5-2

Answer:
It is given that
Finn bought 12 movie tickets. Student tickets cost $4, and adult tickets cost $8. Finn spent a total of $60.
Now,
Let x be the number of student tickets
Let y be the number of adult tickets
So,
x + y = 12 —– (1) [The total number of tickets]
So,
4x + 8y = 60 —– (2) [The total cost of the tickets]
So,
The representation of the above equations in the coordinate plane is:

So,
From the above graph,
The intersection point of the graph is: (9, 3)
Hence, from the above,
We can conclude that
The number of student tickets is: 9
The number of adult tickets is: 3

Question 6.
What value of m gives the system infinitely many solutions? Lesson 5-1
-x + 4y = 32
y = mx + 8
Answer:
The given system of equations are:
-x + 4y = 32
y = mx + 8
So,
4y = x + 32
y = \(\frac{1}{4}\)x + 8 —– (1)
y = mx + 8 ——- (2)
Now,
For the given system of equations to have infinitely many solutions,
The slopes and the y-intercepts of the 2 equations must be equal
So,
m = \(\frac{1}{4}\)
Hence, from the above,
We can conclude that the value of m is: \(\frac{1}{4}\)

Topic 5 MID-TOPIC PERFORMANCE TASK

Perpendicular lines intersect to form right angles. The system of equations below shows perpendicular lines.

PART A
How many solutions does the system have? Explain.
Answer:
The given graph is:

From the given graph,
We can observe that there is only 1 intersection point
Hence, from the above,
We can conclude that the given system of equations has only 1 solution

PART B
Identify the slope and y-intercept of each line. What do you notice about the slopes of the lines?
Answer:
From the given graph,
The system of equations is:
y = \(\frac{3}{4}\)x + 1 —— (1)
y = –\(\frac{4}{3}\)x – 2 —— (2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = \(\frac{3}{4}\), b = 1
For the second equation,
m = –\(\frac{4}{3}\), b = -2
Now,
From the slopes of the 2 equations,
We can observe that the product of the 2 slopes are equal to -1

PART C
What value of m makes the system show perpendicular lines? Explain.
y = \(\frac{1}{2}\)x + 8
y = mx – 6
Answer:
The given system of equations are:
y = \(\frac{1}{2}\)x + 8 —- (1)
y = mx – 6 —— (2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = \(\frac{1}{2}\), b = 8
For the second equation,
m = m, b = -6
We know that,
For the system of equations to be perpendicular,
The product of the slopes must be equal to -1
So,
m₁ . m₂ = -1
So,
\(\frac{1}{2}\)m = -1
m = -2
Hence, from the above,
We ca conclude that the value of m is: -1

Lesson 5.3 Solve Systems by Substitution

Explain It!
Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On-Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On-Time Cabs will cost less.

I can… solve systems of equations using substitution.

A. Do you agree with Jackson? Explain.
Answer:
It is given that
Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On-Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On-Time Cabs will cost less.
Now,
We know that,
For a linear graph,
Quantity 1 (The component of the x-axis) ∝ Quantity 2 (The component of the y-axis)
So,
From the given information,
Slope ∝ Cost
Hence, from the above information,
We can agree with Jackson

B. Which taxi service company should Jackson call? Explain your reasoning.
Answer:
We know that,
The taxi service company must be chosen according to the cost
Hence, from the above,
We can conclude that Jackson should call for an On-Time cabs taxi service since the cost is less

Focus on math practices
Be Precise Can you use the graph to determine the exact number of miles for which the cost of the taxi ride will be the same? Explain.
Answer:
From the given graph,
We can observe that the intersection point is the point where the cost of the taxi ride will be the same
So,
Corresponding to that intersection point, the distance will be calculated for the same cost of the taxi rides
Hence, from the above,
We can conclude that you can use the graph to determine the exact number of miles for which the cost of the taxi ride will be the same

Essential Question When is substitution a useful method for solving systems of equations?
Answer:
The substitution method is most useful for systems of 2 equations in 2 unknowns. The main idea here is that we solve one of the equations for one of the unknowns, and then substitute the result into the other equation.

Try It!

Brandon took a 50-question exam worth a total of 160 points. There were x two-point questions and y five-point questions. How many of each type of question were on the exam?
x + y = 50
2x + 5y = 160
y = _____ – ______
Substitute for y: 2x + 5(_____ – _____) = 160
2x + _____ – ______x = 160
x = ______ two-point questions
Substitute for x: _____ + y = 50
y = _____ five-point questions
Answer:
It is given that
Brandon took a 50-question exam worth a total of 160 points. There were x two-point questions and y five-point questions.
Now,
The given system of equations are:
x + y = 50 —– (1)
2x + 5y = 160 —–(2)
So,
From eq (1),
y = 50 – x
Now,
Substitute y in eq (2)
So,
2x + 5 (50 – x) = 160
2x + 5 (50) – 5x = 160
2x + 250 – 5x = 160
-3x = 160 – 250
-3x = -90
3x = 90
x = \(\frac{90}{3}\)
x = 30
So,
y = 50 – x
y = 50 – 30
y = 20
Hence from the above,
We can conclude that
The number of 2-point questions is: 30
The number of 5-point questions is: 20

Convince Me!
How do you know which equation to choose to solve for one of the variables?
Answer:
The idea of substitution is that if one variable lets you express one variable in terms of the other, you can substitute that expression for the variable in the other equation. That way the second equation only has one variable, and you can solve that

Try It!

Use substitution to solve each system of equations. Explain.
a. y + \(\frac{1}{2}\)x = 3
4y + 2x = -6
Answer:
The given system of equations are:
y + \(\frac{1}{2}\)x = 3 —- (1)
4y + 2x = -6 ——- (2)
So,
From eq (1),
y = 3 – \(\frac{1}{2}\)x
Now,
Substitute y in eq (2)
So,
4 (3 – \(\frac{1}{2}\)x) + 2x = -6
4 (3) – 2x + 2x = -6
12 = -6
Hence, from the above,
We can conclude that there is no solution for the given system of equations

b. y = \(\frac{1}{4}\)x – 2
8y – 2x = -16
Answer:
The given system of equations are:
y = \(\frac{1}{4}\)x – 2 —– (1)
8y – 2x = -16
Now,
Substitute eq (1) in eq (2)
So,
8 (\(\frac{1}{4}\)x – 2) – 2x = -16
2x – 8 (2) – 2x = -16
-16 = -16
16 = 16
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

KEY CONCEPT

Systems of linear equations can be solved algebraically. When one of the equations can be easily solved for one of the variables, you can use substitution to solve the system efficiently.
STEP 1 Solve one of the equations for one of the variables. Then substitute the expression into the other equation and solve.
STEP 2 Solve for the other variable using either equation.

Do You Understand?
Question 1.
Essential Question when is substitution a useful method for solving systems of equations?
Answer:
The substitution method is most useful for systems of 2 equations in 2 unknowns. The main idea here is that we solve one of the equations for one of the unknowns, and then substitute the result into the other equation.

Question 2.
Generalize when using substitution to solve a system of equations, how can you tell when a system has no solution?
Answer:
When a system has no solution or an infinite number of solutions and we attempt to find a single, unique solution using an algebraic method, such as substitution, the variables will cancel out and we will have an equation consisting of only constants. If the equation is untrue then the system has no solution.

Question 3.
Construct Arguments Kavi solved the system of equations as shown. What mistake did Kavi make? What is the correct solution?
3x + 4y = 33
2x + y = 17
y = 17 – 2x
2x + (17 – 2x) = 17
2x + 17 – 2x = 17
2x – 2x + 17 = 17
17 = 17
Infinitely many solutions
Answer:
The given system of equations are:
3x + 4y = 33 —– (1)
2x + y = 17 —– (2)
From eq (2),
y = 17 – 2x
Now,
Substitute y in eq (1)
So,
3x + 4 (17 – 2x) = 33
3x + 4 (17) – 4 (2x) = 33
3x + 68 – 8x = 33
-5x = 33 – 68
-5x = -35
5x = 35
x = \(\frac{35}{5}\)
x = 7
Hence, from the above,
We can conclude that the mistake did by Kavi is the miswriting of eq (1)

Do You Know How?
In 4-6, solve each system using substitution.
Question 4.
y = \(\frac{1}{2}\)x + 4
x – y = 8
Answer:
The given system of equations are:
y = \(\frac{1}{2}\)x + 4 —– (1)
x – y = 8 —– (2)
Now,
From eq (2),
y = x – 8
Substitute y in eq (1)
So,
x – 8 = \(\frac{1}{2}\)x + 4
x – \(\frac{1}{2}\)x = 4 + 8
\(\frac{1}{2}\)x = 12
x = 12 (2)
x = 24
So,
y = x – 8
y = 24 – 8
y = 16
Hence, from the above,
We can conclude that the solution for the given system of equations is: (24, 16)

Question 5.
3.25x – 1.5y = 1.25
13x – y = 10
Answer:
The given system of equations are:
3.25x – 1.5y = 1.25 —- (1)
13x – y = 10 —-(2)
Now,
From eq (2),
y = 13x – 10
Now,
Substitute y in eq (1)
So,
3.25x – 1.5 (13x – 10) = 1.25
3.25x – 1.5 (13x) + 1.5(10) = 1.25
3.25x – 19.5x + 15 = 1.25
-16.25x = -13.75
16.25x = 13.75
x = 0.84
So,
y = 13x – 10
y = 13 (0.84) – 10
y = 0.92
Hence, from the above,
We can conclude that the solution for the given system of equations is: (0.84, 0.92)

Question 6.
y – 0.8x = 0.5
5y – 2.5 = 4x
Answer:
The given system of equations are:
y – 0.8x = 0.5 —- (1)
5y – 2.5 = 4x —— (2)
Now,
From eq (1),
y = 0.8x + 0.5
Now,
Substitute y in eq (2)
So,
5 (0.8x + 0.5) – 2.5 = 4x
5 (0.8x) + 5 (0.5) – 2.5 = 4x
4x + 2.5 – 2.5 = 4x
4x – 4x + 2.5 = 2.5
2.5 = 2.5
Hence, from the above,
We can conclude that the given system of equations has infinitely many solutions

Practice & Problem Solving

Leveled Practice In 7-9, solve the systems of equations.
Question 7.
Pedro has 276 more hits than Ricky. Use substitution to solve the system of equations to find how many hits Pedro, p, and Ricky, r, have each recorded.
p + r = 2,666
p = r + 276

It is given that Pedro has 276 more hits than Ricky
Now,
The given system of equations are:
p + r = 2,666
p = r + 276
Now,
STEP 1 Substitute for p to solve for r.
p + r = 2,666
r + 276 + r = 2,666
2r + 276 = 2,666
2r = 2,390
r = 1,195
STEP 2 Substitute for r to solve for p.
p = r + 276
p = 1,195 + 276
p = 1,471
Hence, from the above,
We can conclude that
Pedro has 1,471 hits, and Ricky has 1,195 hits.

Question 8.
2y + 4.4x = -5
y = -2.2x + 4.5
2 (-2.2x + 4.5) + 4.4x = -5
-4.4x + 9 + 4.4x = -5
9 = -5
Hence, from the above,
We can conclude that
The statement is not true. So, there is no solution.

Question 9.
x + 5y = 0
25y = -5x
x = -5y
25y = -5 (-5y)
25y = 25y
Hence, from the above,
We can conclude that
The statement is true. So, there are infinitely many solutions

Question 10.
On a certain hot summer day, 481 people used the public swimming pool. The daily prices are $1.25 for children and $2.25 for adults. The receipts for admission totaled $865.25. How many children and how many adults swam at the public pool that day?
Answer:
It is given that
On a certain hot summer day, 481 people used the public swimming pool. The daily prices are $1.25 for children and $2.25 for adults. The receipts for admission totaled $865.25.
Now,
Let x be the number of children
Let y be the number of adults
So,
x + y = 481 —– (1) [The number of people that used the public swimming pool]
1.25x + 2.25y = 865.25 —– (2) [The receipts for admission]
Now,
From eq (1),
y = 481 – x
Substitute y in eq (2)
So,
1.25x + 2.25 (481 – x) = 865.25
1.25x + 2.25 (481) – 2.25 (x) = 865.25
1.25x + 1,082.25 – 2.25x = 865.25
-x = -217
x = 217
Now,
y = 481 – x
y = 481 – 217
y = 264
Hence,from the above,
We can conclude that
The number of children that swam at the pool is: 217
The number of adults that swam at the pool is: 264

Question 11.
Construct Arguments Tim incorrectly says that the solution of the system of equations is x = -9, y = -4.
6x – 2y = -6
11 = y – 5x
a. What is the correct solution?
Answer:
The given system of equations are:
6x – 2y = -6 —– (1)
11 = y – 5x —— (2)
Now,
From eq (2),
y = 5x + 11
Substitute y in eq (1)
So,
6x – 2 (5x + 11) = -6
6x – 2 (5x) – 2 (11) = -6
6x – 10x – 22 = -6
-4x = 16
x = –\(\frac{16}{4}\)
x = -4
Now,
y = 5x + 11
y = 5 (-4) + 11
y = -20 + 11
y = -9
Hence, from the above,
We can conclude that the correct solution for the given system of equations is: (-4, -9)

b. What error might Tim have made?
Answer:
It is given that
Tim incorrectly says that the solution of the system of equations is x = -9, y = -4.
But, from part (a),
We can observe that x = -4, y = -9
Hence, from the above,
We can conclude that the error that Tim has made is the reversal of the values of x and y in the solution

Question 12.
The number of water bottles, y, filled in x minutes by each of two machines is given by the equations below. Use substitution to determine if there is a point at which the machines will have filled the same number of bottles.
160x + 2y = 50
y + 80x = 50
Answer:
The given system of equations are:
160x + 2y = 50 —— (1)
y + 80x = 50 ——- (2)
Now,
From eq (2),
y = 50 – 80x
Now,
Substitute y in eq (1)
So,
160x + 2 (50 – 80x) = 50
160x + 2 (50) – 2 (80x) = 50
160x + 100 – 160x = 50
100 = 50
Hence, from the above,
We can conclude that at any point, the machines will not have filled the same number of bottles

Question 13.
a. Use substitution to solve the system below.
x = 8y – 4
x + 8y = 6
Answer:
The given system of equations are:
x = 8y – 4 —– (1)
x + 8y = 6 —– (2)
Now,
Substitute eq (1) in eq (2)
So,
8y – 4 + 8y = 6
16y – 4 = 6
16y = 10
y = \(\frac{16}{10}\)
y = 1.6
Now,
x = 8y – 4
x = 8 (1.6) – 4
x = 12.8 – 4
x = 8.8
Hence, from the above,
We can conclude that

b. Reasoning Which expression would be easier to substitute into the other equation in order to solve the problem? Explain.
Answer:
From part (a),
The given system of equations are:
x = 8y – 4 —– (1)
x + 8y = 6 —– (2)
Hence,
From the given equations,
In order to reduce the number of steps,
Eq (1) would be easier to substitute into the other equation in order to solve the problem

Question 14.
The perimeter of a frame is 36 inches. The length is 2 inches greater than the width. What are the dimensions of the frame?

Answer:
The given frame is:

From the given frame,
We can observe that it is in the form of a rectangle
Now,
We know that,
The perimeter of a rectangle = 2 (Length + Width)
So,
36 = 2 (W + 2 + W)
36 = 2 (2W + 2)
2W + 2 = \(\frac{36}{2}\)
2W + 2 = 18
2W = 18 – 2
2W = 16
W = \(\frac{16}{2}\)
W = 8
So,
L = W + 2
L = 8 + 2
L = 10
Hence, from the above,
We can conclude that the dimensions of the frame are:
The length of the frame is: 10 inches
The width of the frame is: 8 inches

Question 15.
Higher-Order Thinking The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at $6.50. Let s represent the number of student tickets sold at $3.50 each.
a. If all 500 seats are filled for a performance, how many of each type of ticket must have been sold for the members to raise exactly $2,050?
Answer:
It is given that
The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at $6.50. Let s represent the number of student tickets sold at $3.50 each.
Now,
d + s = 500 —– (1) [The total number of people]
6.50d + 3.50s = 2,050 —– (2) [The total expenses]
So,
From eq (1),
d = 500 – s
Now,
Substitute d in eq (1)
So,
6.50 (500 – s) + 3.50s = 2,050
6.50 (500) – 6.50s + 3.50s = 2,050
3,250 – 3s = 2,050
-3s =-1,200
3s = 1,200
s = \(\frac{1,200}{3}\)
s = 400
So,
d = 500 – s
d = 500 – 400
d = 100
Hence, from the above,
We can conclude that
The number of student tickets is: 400
The number of adult tickets is: 100

b. At one performance there were three times as many student tickets sold as adult tickets. If there were 480 tickets sold at that performance, how much below the goal of $2,050 did ticket sales fall?
Answer:
It is given that there were three times as many student tickets as adult tickets
So,
s = 3d
Now,
d + s = 480 —– (1)
So,
d + 3d = 480
4d = 480
d = \(\frac{480}{4}\)
d = 120
So,
s = 3 (120)
s = 360
Now,
The equation for the total expenses is:
6.50d + 3.50s = 6.50 (120) + 3.50 (360)
= 780 + 1,260
= 2,040
Now,
The fall in the ticket sales = $2,050 – $2,040
= $10
Hence, from the above,
We can conclude that there is a fall of $10 to reach the goal of ticket sales of $2,050

Assessment Practice
Question 16.
What statements are true about the solution of the system?
y = 145 – 5x
0.1y + 0.5x = 14.5
☐ There are infinitely many solutions.
☐ (20, 45) is a solution.
☐ (10, 95) is a solution.
☐ There is no solution.
☐ There is more than one solution.
Answer:
Let the given options be named as A, B, C, D, and E
Now,
The given system of equations are:
y = 145 – 5x —- (1)
0.1y + 0.5x = 14.5 —- (2)
Now,
Substitute eq (1) in eq (2)
So,
0.1 (145 – 5x) + 0.5x = 14.5
0.1 (145) – 0.1 (5x) + 0.5x = 14.5
14.5 – 0.5x + 0.5x = 14.5
14.5 – 0.5x = 14.5 – 0.5x
So,
We can say that there are infinitely many solutions for the given system of equations
Hence, from the above,
We can conclude that option A matches with the solution for the given system of equations

Question 17.
At an animal shelter, the number of dog adoptions one weekend was 10 less than 3 times the number of cat adoptions. The number of cat adoptions plus twice the number of dog adoptions was 8. How many cats and how many dogs were adopted that weekend?
Answer:
It is given that
At an animal shelter, the number of dog adoptions one weekend was 10 less than 3 times the number of cat adoptions. The number of cat adoptions plus twice the number of dog adoptions was 8
Now,
Let the cat adoptions be x
Let the dog adoptions be y
So,
y = 3x – 10 —– (1)
x + 2y = 8 ——- (2)
Now,
Substitute eq (1) in eq (2)
So,
x + 2 (3x – 10) = 8
x – 2 (10) + 2 (3x) = 8
x – 20 + 6x = 8
7x = 28
x = \(\frac{28}{4}\)
x = 7
Now,
y = 3x – 10
y = 3 (7) – 10
y = 21 – 10
y = 11
Hence, from the above,
We can conclude that
The number of dogs adopted that weekend is: 11
The number of cats adopted that weekend is: 7

Lesson 5.4 Solve Systems by Elimination

Solve & Discuss It!
A list of expressions is written on the board. How can you make a list of fewer expressions that has the same combined value as those shown on the board? Write the expressions and explain your reasoning.

I can… solve systems of equations using elimination.
Answer:

Look for Relationships
How can you use what you know about combining like terms to make your list?
Answer:
Like terms are mathematical terms that have the exact same variables and exponents, but they can have different coefficients. Combining like terms will simplify a math problem and is also the proper form for writing a polynomial. To combine like terms, just add the coefficients of each like term

Focus on math practices
Reasoning Two expressions have a sum of 0. What must be true of the expressions?
Answer:
It is given that two expressions have a sum of 0
Hence,
From the given information,
We can conclude that the expressions are the same but they are of the opposite sign

Essential Question
How are the properties of equality used to solve systems of linear equations?
Answer:
Basically, you can add or subtract anything you wish to both sides of any inequality, and you can multiply or divide, too, by any positive number, to get a simpler inequality, on the way to solving for the variable

Try It!

Use elimination to solve the system of equations.
2r + 3s = 14
6r – 3s = 6
The solution is r = ______, s = _____.

Answer:
The given system of equations are:
2r + 3s = 14 —- (1)
6r – s = 6 —– (2)
So,

Hence, from the above,
We can conclude that the solutions for the given system of equations are:
r = \(\frac{25}{4}\) and s = \(\frac{1}{2}\)

Convince Me!
What must be true about a system of equations for a term to be eliminated by adding or subtracting?
Answer:
In the elimination method, you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

Try It!

Use elimination to solve the system of equations.
3x – 5y = -9
x + 2y = 8
Answer:
The given system of equations are:
3x – 5y = -9
x + 2y = 8
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (2, 3)

KEY CONCEPT

You can apply the properties of equality to solve systems of linear equations algebraically by eliminating a variable. Elimination is an efficient method when:
• like variable terms have the same or opposite coefficients.
• one or both equations can be multiplied so that like variable terms have the same or opposite coefficients.

Do You Understand?
Question 1.
Essential Question How are the properties of equality used to solve systems of linear equations?
Answer:
Basically, you can add or subtract anything you wish to both sides of any inequality, and you can multiply or divide, too, by any positive number, to get a simpler inequality, on the way to solving for the variable

Question 2.
How is solving a system of equations algebraically similar to solving the system by graphing? How is it different?
Answer:
To solve a system of linear equations graphically we graph both equations in the same coordinate system. The solution to the system will be at the point where the two lines intersect.

Question 3.
Construct Arguments Consider the system of equations. Would you solve this system by substitution or by elimination? Explain.
1\(\frac{3}{4}\)x + y = 2\(\frac{3}{16}\)
\(\frac{1}{4}\)x – y = -1\(\frac{11}{16}\)
Answer:
The given system of equations are:
1\(\frac{3}{4}\)x + y = 2\(\frac{3}{16}\)
\(\frac{1}{4}\)x – y = -1\(\frac{11}{16}\)
Now,
From the above equations,
We can observe that we can eliminate y easily by using the elimination method
Hence, from the above,
We can conclude that we would solve the given system of equations by using the elimination method

Do You Know How?
In 4–6, solve each system of equations by using elimination.
Question 4.
y – x = 28
y + x = 156
Answer:
The given system of equations are:
y – x = 28 —- (1)
y + x = 156 —- (2)
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (64, 92)

Question 5.
3c + 6d = 18
6C – 2d = 22
Answer:
The given system of equations are:
3c + 6d = 18 — (1)
6c – 2d = 22 —– (2)
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (4, 1)

Question 6.
7x + 14y = 28
5x + 10y = 20
Answer:
The given system of equations are:
7x + 14y = 28 —– (1)
5x + 10y = 20 —– (2)
So,

Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

Practice & Problem Solving

Question 7.
Leveled Practice Solve the system of equations using elimination.
2x – 2y = 4
2x + y = 11
Multiply the first equation by ______.

Answer:
The given system of equations are:
2x – 2y = 4 —– (1)
2x + y = 11 —– (2)
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (\(\frac{13}{3}\), \(\frac{7}{3}\))

Question 8.
Solve the system of equations using elimination.
2y – 5x = -2
3y + 2x = 35
Answer:
The given system of equations are:
2y – 5x = -2 —- (1)
3y + 2x = 35 —- (2)
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (4, 9)

Question 9.
If you add Natalie’s age and Frankie’s age, the result is 44. If you add Frankie’s age to 3 times Natalie’s age, the result is 70. Write and solve a system of equations using elimination to find their ages.
Answer:
It is given that
If you add Natalie’s age and Frankie’s age, the result is 44. If you add Frankie’s age to 3 times Natalie’s age, the result is 70.
Now,
Let Natalie’s age be n
Let Frankie’s age be f
So,
From the given information,
The system of equations that can be formed is:
n + f = 44 —- (1)
f + 3n = 70 —- (2)
Now,
Eq (1) – Eq (2)
So,

Hence, from the above,
We can conclude that
The age of Natalie is: 13 years
The age of Frankie is: 31 years

Question 10.
If possible, use elimination to solve the system of equations.
5x + 10y = 7
4x + 8y = 3
Answer:
The given system of equations are:
5x + 10y = 7 —- (1)
4x + 8y = 3 —— (2)
So,

Hence, from the above,
We can conclude that the given system of equations has no solution

Question 11.
At a basketball game, a team made 56 successful shots. They were a combination of 1- and 2-point shots. The team scored 94 points in all. Use elimination to solve the system of equations to find the number of each type of shot.
x + y = 56
x + 2y = 94
Answer:
It is given that
At a basketball game, a team made 56 successful shots. They were a combination of 1- and 2-point shots. The team scored 94 points in all
Now,
The given system of equations that represent the given situation is:
x + y = 56 —- (1)
x + 2y = 94 —- (2)
Now,
Eq (1) – Eq (2)
So,

Hence, from the above,
We can conclude that
The number of 1-point shots is: 18
The number of 2-point shots is: 38

Question 12.
Two trains, Train A and Train B, weigh a total of 312 tons. Train A is heavier than Train B. The difference in their weights is 170 tons. Use elimination to solve the system of equations to find the weight of each train.
a + b = 312
a – b= 170

Answer:
It is given that
Two trains, Train A and Train B weigh a total of 312 tons. Train A is heavier than Train B. The difference in their weights is 170 tons.
Now,
The system of equations given that describes the given situation is:
a + b = 312 —– (1)
a – b = 170 —— (2)
Now,
Eq (1) – Eq (2)
So,

Hence, from the above,
We can conclude that
The weight of Train A is: 241 tons
The weight of Train B is: 71 tons

Question 13.
A deli offers two platters of sandwiches. Platter A has 2 roast beef sandwiches and 3 turkey sandwiches. Platter B has 3 roast beef sandwiches and 2 turkey sandwiches.

a. Model with Math Write a system of equations to represent the situation.
Answer:
It is given that
A deli offers two platters of sandwiches. Platter A has 2 roast beef sandwiches and 3 turkey sandwiches. Platter B has 3 roast beef sandwiches and 2 turkey sandwiches.
Now,
Let a piece of roast beef sandwich be r
Let a piece of turkey beef sandwich be t
So,
In plate A,
The number of sandwiches is: 2r + 3t
In plate B,
The number of sandwiches is: 3r + 2t
Now,
From the above figure,
The total cost of plates A and B are:
2r + 3t = 31
3r + 2t = 29
Hence, from the above,
We can conclude that the system of equations that represent the given situation is:
2r + 3t = 31
3r + 2t = 29

b. What is the cost of each sandwich?
Answer:
From part (a),
The system of equations are:
2r + 3t = 31 —– (1)
3r + 2t = 29 —– (2)
So,

Hence, from the above,
We can conclude that
The cost of each roast beef sandwich is: $5
The cost of each turkey beef sandwich is: $7

Question 14.
Consider the system of equations.
x – 3.1y = 11.5
-x + 3.5y = -13.5
a. Solve the system by elimination.
Answer:
The given system of equations are:
x – 3.1y = 11.5 —- (1)
-x + 3.5y = -13.5 —— (2)
Now,
Eq (1) – Eq (2)
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (-4, -5)

b. If you solved this equation by substitution instead, what would the solution be? Explain.
Answer:
By using the substitution method,
The solution for the given system of equations is:

Hence, from the above,
We can conclude that the solution for the substitution method is the same as the solution for the elimination method

Question 15.
Higher-Order Thinking Determine the number of solutions for this system of equations by inspection only. Explain.
3x + 4y = 17
21x + 28y = 109
Answer:
The given system of equations are:
3x + 4y = 17 —– (1)
21x + 28y = 109 —— (2)
So,

Hence, from the above,
We can conclude that there is no solution for the given system of equations

Assessment Practice
Question 16.
Four times a number r plus half a number s equals 12. Twice the number r plus one fourth of the number s equals 8. What are the two numbers?
Answer:
It is given that
Four times a number r plus half a number s equals 12. Twice the number r plus one-fourth of the number s equals 8.
So,
4r + \(\frac{1}{2}\)s = 12 —- (1)
2r + \(\frac{1}{4}\)s = 8 —- (2)
So,

Hence, from the above,
We can conclude that the values for the 2 numbers are not possible

Question 17.
Solve the system of equations.
3m + 3n = 36
8m – 5n = 31
Answer:
The given system of equations are:
3m + 3n = 36 —- (1)
8m – 5n = 31 —– (2)
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (7, 5)

3-ACT MATH

3-Act Mathematical Modeling: Ups and Downs

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

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

Question 3.
Make a prediction to answer this Main Question.
The person who wins took the ______.
Answer:

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

ACT 2
Question 5.
What information in this situation would be helpful to know? How would you use that information?
Answer:

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

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

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

Answer:

ACT 3
Question 9.
Write the answer you saw in the video.

Answer:

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

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

Answer:

ACT 3 Extension
Reflect
Question 12.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 13.
Reason Abstractly A classmate solved the problem using equations with independent variable a and dependent variable b. What do these variables represent in the situation?
Answer:

SEQUEL
Question 14.
Generalize Write an equation or inequality to represent all numbers of flights for which the elevator is faster.

Answer:

Topic 5 REVIEW

Topic Essential Question
What does it mean to solve a system of linear equations?
Answer:
The solution to a system of linear equations is the point at which the lines representing the linear equations intersect. Two lines in the XY -plane can intersect once, never intersect, or completely overlap.

Vocabulary Review

Complete each definition and then provide an example of each vocabulary word.
Vocabulary
solution of a system of linear equations
system of linear equations

Answer:

Use Vocabulary in Writing
Describe how you can find the number of solutions of two or more equations by using the slope and the y-intercept. Use vocabulary terms in your description.
Answer:
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions

Concepts and Skills Review

Lesson 5.1 Estimate Solutions by Inspection

Quick Review
The slopes and y-intercepts of the linear equations in a system determine the relationship between the lines and the number of solutions.

Example
How many solutions does the system of equations have? Explain.
Answer:
y + 2x = 6
y – 8 = -2x
Write each equation in slope-intercept form.
y = -2x + 6
y = -2x + 8
Identify the slope and y-intercept of each equation.
For the equation, y = -2x + 6, the slope is –2 and the y-intercept is 6.
For the equation, y = -2x + 8, the slope is -2 and the y-intercept is 8.
The equations have the same slope but different y-intercepts, so the system has no solution.

Practice
Determine whether the system of equations has one solution, no solution, or infinitely many solutions.
Question 1.
y – 13 = 5x
y – 5x = 12
Answer:
The given system of equations are:
y – 13 = 5x
y – 5x = 12
So,
The required system of equations are:
y = 5x + 13 —– (1)
y = 5x + 12 —– (2)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = 5, b = 13
For the second equation,
m = 5, b = 12
We know that,
When slopes are the same, the lines may be either parallel (different y-intercepts)
There are no solutions
Hence, from the above,
We can conclude that there is no solution for the given system of equations

Question 2.
y = 2x + 10
3y – 6x = 30
Answer:
The given system of equations are:
y = 2x + 10 —– (1)
3y – 6x = 30 —– (2)
So,
Divide eq (2) by 3
y – 2x = 10
So,
y = 2x + 10 —– (3)
Now,
Compare the above equations with
y = mx + b
So,
For the first equation,
m = 2, b = 10
For the third equation,
m = 2, b = 10
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

Question 3.
-3x + \(\frac{1}{3}\)y = 12
2y = 18x + 72
Answer:
The given system of equations are:
-3x + \(\frac{1}{3}\)y = 12 —– (1)
2y = 18x + 72 —- (2)
So,
Multiply eq (1) with 3
-9x + y = 36
y = 9x + 36 —– (3)
Now,
Divide eq (2) with 2
So,
y = 9x + 36 —– (4)
Now,
Compare the above equations with
y = mx + b
So,
From the third equation,
m = 9, b = 36
From the fourth equation,
m = 9, b = 36
We know that,
If the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
So, there are infinitely many solutions
Hence, from the above,
We can conclude that there are infinitely many solutions for the given system of equations

Question 4.
y – \(\frac{1}{4}\)x = -1
y – 2 = 4x
Answer:
The given system of equations are:
y – \(\frac{1}{4}\)x = -1 —- (1)
y – 2 = 4x —— (2)
Now,
y = \(\frac{1}{4}\)x – 1 —- (3)
y = 4x + 2 —- (4)
Compare the above equations with
y = mx + b
So,
From the third equation,
m = \(\frac{1}{4}\), b = -1
From the fourth equation,
m = 4, b = 2
We know that,
When slopes and y-intercepts are different, the two lines intersect at one point,
There is only 1 solution.
Hence, from the above,
We can conclude that there is only 1 solution for the given system of equations

Question 5.
Michael and Ashley each buy x pounds of turkey and y pounds of ham. Turkey costs $3 per pound at Store A and $4.50 per pound at Store B. Ham costs $4 per pound at Store A and $6 per pound at Store B. Michael spends $18 at Store A, and Ashley spends $27 at Store B. Could Michael and Ashley have bought the same amount of turkey and ham? Explain.
Answer:
It is given that
Michael and Ashley each buy x pounds of turkey and y pounds of ham. Turkey costs $3 per pound at Store A and $4.50 per pound at Store B. Ham costs $4 per pound at Store A and $6 per pound at Store B. Michael spends $18 at Store A, and Ashley spends $27 at Store B.
Now,
Take the number of pounds for the turkey to be x and that for the ham to be y
For store A where Michael spent $18,
Turkey cost $3 per pound —- 3x
Ham cost $4 per pound——4y
So,
The equation for cost will be;
3x + 4y = 18
Now,
For store B where Ashley spent $27
Turkey cost $4.5 per pound
Ham cost $6 per pound
So,
The equation for cost is:
4.5x + 6y = 27
So,
The two equations are;
3x + 4y = 18 —— (1)
4.5x + 6y = 27 —– (2)
Now,
Divide eq (2) with 3
So,
1.5x + 2y = 9 —- (3)
Multiply the above equation with 2
So,
3x + 4y = 18 — (4)
Now,
Compare eq (1) and eq (4) with
y = mx + b
So,
From the eq (1),
m = 3, b = 4
From the eq (2),
m = 3, b = 4
From the above,
We can observe that the slopes and the y-intercepts are equal
So,
Both the equations are in the same line
Hence, from the above,
We can conclude that Michael and Ashley bought the same amount of Turkey and Ham

Lesson 5.2 Solve Systems by Graphing

Quick Review
Systems of equations can be solved by looking at their graphs. A system with one solution has one point of intersection. A system with infinitely many solutions has infinite points of intersection. A system with no solution has no points of intersection.

Example
Graph the system and determine its solution.
y = x + 4
y = -2x + 1
Answer:
Graph each equation in the system on the same coordinate plane.

The point of intersection is (-1, 3). This means the solution to the system is (-1, 3).

Practice
Graph each system and find the solution(s).
Question 1.
y = \(\frac{1}{2}\)x + 1
-2x + 4y = 4

Answer:
The given system of equations are:
y = \(\frac{1}{2}\)x + 1
-2x + 4y = 4
So,
The representation of the given system of equations in a coordinate plane is:

Hence, from the above,
We can conclude that the given system of equations has infinitely many solutions

Question 2.
y = -x – 3
y + x = 2

Answer:
The given system of equations are:
y = -x – 3
y + x = 2
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that there is no solution for the given system of equations

Question 3.
2y = 6x + 4
y = -2x + 2

Answer:
The given system of equations are:
2y = 6x + 4
y = -2x + 2
So,
The representation of the given system of equations in the coordinate plane is:

Hence, from the above,
We can conclude that the solution for the given system of equations is: (0, 2)

Lesson 5.3 Solve Systems by Substitution

Quick Review
To solve a system by substitution, write one equation for a variable in terms of the other. Substitute the expression into the other equation and solve. If the result is false, the system has no solution. If true, it has infinitely many solutions. If the result is a value, substitute to solve for the other variable.

Example
Use substitution to solve the system.
y = x + 1
y = 5x – 3
Answer:
Substitute x + 1 for y in the second equation.
(x + 1) = 5x – 3
4 = 4x
1 = x
Substitute 1 for x in the first equation.
y = (1) + 1 = 2
The solution is x = 1, y = 2.

Practice
Use substitution to solve each system.
Question 1.
-3y = -2x – 1
y = x – 1
Answer:
The given system of equations are:
-3y = -2x – 1 —– (1)
y = x – 1 —- (2)
Now,
Substitute eq (2) in eq (1)
So,
-3 (x – 1) = -2x – 1
-3 (x) + 3 (1) = -2x – 1
-3x + 3 = -2x – 1
-3x + 2x = -1 – 3
-x = -4
x = 4
So,
y = x – 1
y = 4 – 1
y = 3
Hence, from the above,
We can conclude that the solution for the given system of equations is: (4, 3)

Question 2.
y = 5x + 2
2y – 4 = 10x
Answer:
The given system of equations are:
y = 5x + 2 —- (1)
2y – 4 = 10 —– (2)
Now,
Substitute eq (1) in eq (2)
So,
2 (5x + 2) – 4 = 10
2 (5x) + 2 (2) – 4 = 10
10x = 10
x = 1
So,
y = 5x + 2
y = 5 + 2
y = 7
Hence, from the above,
We can conclude that the solution for the given system of equations is: (1, 7)

Question 3.
2y – 8 = 6x
y = 3x + 2
Answer:
The given system of equations are:
2y – 8 = 6x —- (1)
y = 3x + 2 —– (2)
Now,
Substitute eq (2) in eq (1)
So,
2 (3x + 2) – 8 = 6x
2 (3x) + 2 (2) – 8 = 6x
6x + 4 – 8 = 6x
4 = 8
Hence, from the above,
We can conclude that there is no solution for the given system of equations

Question 4.
2y – 2 = 4x
y = -x + 4
Answer:
The given system of equations are:
2y – 2 = 4x —– (1)
y = -x + 4 —– (2)
Now,
Substitute eq (2) in eq (1)
So,
2 (-x + 4) – 2 = 4x
2 (-x) + 2 (4) – 2 = 4x
-2x + 8 – 2 = 4x
4x + 2x = 6
6x = 6
x = 1
So,
y = -x + 4
y = -1 + 4
y = 3
Hence, from the above,
We can conclude that the solution for the given system of equations is: (1, 3)

Lesson 5.4 Solve Systems by Elimination

Quick Review
To solve a system by elimination, multiply one or both equations to make opposite terms. Add (or subtract) the equations to eliminate one variable. Substitute to solve for the other variable.

Example
Use elimination to solve the system.
2x – 9y = -5
4x – 6y = 2
Multiply the first equation by -2.Then add.

y = 1
Substitute 1 for y in the first equation.
2x – 9(1) = -5
2x – 9 = -5
2x = 4
x = 2
The solution is x = 2, y = 1.

Practice
Use elimination to solve each system.
Question 1.
-2x + 2y = 2
4x – 4y = 4
Answer:
The given system of equations are:
-2x + 2y = 2 —– (1)
4x – 4y = 4 ——- (2)
So,

Hence, from the above,
We can conclude that the given system of equations has no solution

Question 2.
4x + 6y = 40
-2x + y = 4
Answer:
The given system of equations are:
4x + 6y = 40
-2x + y = 4
So,

Hence, from the above,
We can conclude that the solution for the given system of equations is: (1, 6)

Question 3.
A customer at a concession stand bought 2 boxes of popcorn and 3 drinks for $12. Another customer bought 3 boxes of popcorn and 5 drinks for $19. How much does a box of popcorn cost? How much does a drink cost?
Answer:
It is given that
A customer at a concession stands bought 2 boxes of popcorn and 3 drinks for $12. Another customer bought 3 boxes of popcorn and 5 drinks for $19
Now,
Let each box of popcorn be p
Let each drink be d
So,
For Customer A,
2p + 3d = $12 —– (1)
For Customer B,
3p + 5d = $19 —– (2)
Now,

Hence, from the above,
We can conclude that
The cost of each popcorn box is: $3
The cost of each drink is: $2

Topic 6 Fluency Practice

Pathfinder
Shade a path from START to FINISH. Follow the solutions to the equations from least to greatest. You can only move up, down, right, or left.
I can… solve multistep equations using the Distributive Property.