REVIEWTOPIC

2Topic Essential Question?

Vocabulary ReviewComplete each definition and provide an example of each vocabulary word.

Vocabulary slope of a line y-intercept slope-intercept form x-intercept

Definition Example

1. The change in y divided by the change in x is the .

2. The point on the graph where the line crosses the y-axis is the of a line.

3. The of a line is y = mx + b. The variable m in the equation stands for the . The variable b in the equation stands for the .

Use Vocabulary in WritingPaddle boats rent for a fee of $25, plus an additional $12 per hour. What equation, in y = mx + b form, represents the cost to rent a paddle boat for x hours? Explain how you write the equation. Use vocabulary words in your explanation.

How can you analyze connections between linear equations and use them to solve problems?

Topic 2 Topic Review 147

Solve Equations with Variables on Both Sides

LESSON 2-1 Combine Like Terms to Solve Equations

Quick Review

You can use variables to represent unknown quantities. To solve an equation, collect like terms to get one variable on one side of the equation. Then use inverse operations and properties of equality to solve the equation.

Example

Solve 5x + 0.45x = 49.05 for x.

5x + 0.45x = 49.05

5.45x = 49.05

5.45x5.45 = 49.05

5.45

x = 9

Practice

Solve each equation for x.

1. 2x + 6x = 1,000

2. 214x + 1

2x = 44

3. -2.3x - 4.2x = -66.3

4. Javier bought a microwave for $105. The cost was 30% off the original price. What was the price of the microwave before the sale?

Quick Review

If two quantities represent equal amounts and have the same variables, you can set the expressions equal to each other. Collect all the variables on one side of the equation and all the constants on the other side. Then use inverse operations and properties of equality to solve the equation.

Example

Solve 2x + 21 = 7x + 6 for x.

2x + 21 = 7x + 6

21 = 5x + 6

15 = 5x

x = 3

Practice

Solve each equation for x.

1. 3x + 9x = 6x + 42

2. 43x + 23x = 1

3x + 5

3. 9x - 5x + 18 = 2x + 34

4. Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week. After how many weeks will Megan and Connor have saved the same amount?

Concepts and Skills Review

LESSON 2-2

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LESSON 2-3 Solve Multistep Equations

LESSON 2-4 Equations with No Solutions or Infinitely Many Solutions

Quick Review

When solving multistep equations, sometimes the Distributive Property is used before you collect like terms. Sometimes like terms are collected, and then you use the Distributive Property.

Example

Solve 8x + 2 = 2x + 4(x + 3) for x.

First, distribute the 4. Then, combine like terms. Finally, use properties of equality to solve for x.

8x + 2 = 2x + 4x + 12

8x + 2 = 6x + 12

8x = 6x + 10

2x = 10

x = 5

Practice

Solve each equation for x.

1. 4(x + 4) + 2x = 52

2. 8(2x + 3x + 2) = -4x + 124

3. Justin bought a calculator and a binder that were both 15% off the original price. The original price of the binder was $6.20. Justin spent a total of $107.27. What was the original price of the calculator?

Quick Review

When solving an equation results in a statement that is always true, there are infinitely many solutions. When solving an equation produces a false statement, there are no solutions. When solving an equation gives one value for a variable, there is one solution.

Example

How many solutions does the equation 6x + 9 = 2x + 4 + 4x + 5 have?

First, solve the equation.

6x + 9 = 2x + 4 + 4x + 5

6x + 9 = 6x + 9

9 = 9

Because 9 = 9 is alwyas a true statement, the equation has infinitely many solutions.

Practice

How many solutions does each equation have?

1. x + 5.5 + 8 = 5x - 13.5 - 4x

2. 4(12x + 3) = 3x + 12 - x

3. 2(6x + 9 - 3x) = 5x + 21

4. The weight of Abe’s dog can be found using the expression 2(x + 3), where x is the number of weeks. The weight of Karen’s dog can be found using the expression 3(x + 1), where x is the number of weeks. Will the dogs ever be the same weight? Explain.

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LESSON 2-5 Compare Proportional Relationships

Quick Review

To compare proportional relationships, compare the rate of change or find the unit rate.

Example

The graph shows the rate at which Rob jogs. Emily’s jogging rate is represented by the equation y = 8x, where x is the number of miles and y is the number of minutes. At these rates, who will finish an 8-mile race first?

0 1 2 3 4 5 6 7 8 90

6

9

12

15

18

21

24

27

3

Min

ute

s

Miles

y

x

Emily’s unit rate is y = 8(1) = 8 minutes per mile.

The point (1, 6) represents Rob’s unit rate of 6 minutes per mile.

Rob’s unit rate is less than Emily’s rate, so Rob will finish an 8-mile race first.

Practice

1. Two trains are traveling at a constant rate. Find the rate of each train. Which train is traveling at the faster rate?

Time (h) 2 3 4 5 6

Distance (mi) 50 75 100 125 150

Train A

Train B

0 1 2 3 4 5 6 7 8 90

20

30

40

50

60

70

80

90

10

Mile

s

Hours

y

x

2. A 16-ounce bottle of water from Store A costs $1.28. The cost in dollars, y, of a bottle of water from Store B is represented by the equation y = 0.07x, where x is the number of ounces. What is the cost per ounce of water at each store? Which store’s bottle of water costs less per ounce?

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LESSON 2-6 Connect Proportional Relationships and Slope

LESSON 2-7 Analyze Linear Equations: y = mx

Quick Review

The slope of a line in a proportional relationship is the same as the unit rate and the constant of proportionality.

Example

The graph shows the number of miles a person walked at a constant speed. Find the slope of the line.

0 1 2 3 4 50

203040506070

10

Min

ute

s

Miles

y

x

slope =y2 - y1x2 - x1

= 60 - 304 - 2 = 30

2 = 15

Practice

1. The graph shows the proportions of blue paint and yellow paint that Briana mixes to make green paint. What is the slope of the line? Tell what it means in the problem situation.

0 1 2 3 4 5 6 7 8 90

23456789

1

Yel

low

Pai

nt

Blue Paint

y

x

Quick Review

A proportional relationship can be represented by an equation in the form y = mx, where m is the slope.

Example

Graph the line y = 2x.

Plot a point at (0, 0). Then use the slope to plot the next point.

y

x2

2

O

Practice

A mixture of nuts contains 1 cup of walnuts for every 3 cups of peanuts.

1. Write a linear equation that represents the relationship between peanuts, x, and walnuts, y.

2. Graph the line.

0 1 2 30

23

1

Wal

nu

ts (c

)

Peanuts (c)

y

x

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LESSON 2-8 Understand the y-Intercept of a Line

Quick Review

The y-intercept is the y-coordinate of the point where a line crosses the y-axis. The y-intercept of a proportional relationship is 0.

Example

What is the y-intercept of the line?

y

x2

2

22

22

O

The y-intercept is 0.

Practice

The equation y = 5 + 0.5x represents the cost of getting a car wash and using the vacuum for x minutes.

0 1 2 3 4 5 6 70

234567

1

Co

st ($

)

Number of Minutes

y

x

1. What is the y-intercept?

2. What does the y-intercept represent?

Analyze Linear Equations: y = mx + b

Quick Review

An equation in the form y = mx + b, where b 3 0, has a slope of m and a y-intercept of b. This form is called the slope-intercept form. There is not a proportional relationship between x and y in these cases.

Example

What is the equation of the line?

y

x2

2

22

22 O

Since m = 2 and b = -3, the equation is y = 2x - 3.

Practice

1. Graph the line with the equation y = 12x - 1.

y

x2

2

22

22 O

2. What is the equation of the line?

y

x2

2

22

22 O

LESSON 2-9

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