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Name

CHAPTER

Date

Class

Reteach

11-3 Solving Equations with Variables on Both Sides

Some equations have like terms that are on opposite sides of the equal sign. To solve these equations, first identify the like term with the lesser coefficient. 2x 25 3x 2x and 3x are like terms. Since 3 2, 3x is the term with the 2x lesser coefficient. Move 3x to the 2x 3x opposite side of the equation by adding 5x 5x 3x to each side of the equation.

5

25 25 25 5

25 5

3x 3x

3x

x

Complete the steps to solve the equation. 1. 5x 16 9x . 2. 3a 1 15 4a .

The like terms are and The like term with the lesser coefficient is . 9x 5x 16 5x 16 9x 16

16 4x

The like terms are and The like term with the lesser coefficient is . 3a 1 15 4a 3a 1 15 4a 1 15 7a 1 7a

7a 14

15

x

a Solve. 3. 8d 7d 13 4. 4p 16 4p 5. 3n 25 2n

6. 9c

6

2c

15

7. 16

6k

20

2k

8. 2h

8

2

h

Copyright © by Holt, Rinehart and Winston. All rights reserved.

30

Holt Middle School Math Course 2

Name

LESSON

Date

Class

Practice B

11-3 Solving Equations with Variables on Both Sides

Group the terms with the variables on one side of the equal sign and simplify. Do not solve. 1. 10t 6t 24 2. 6x 32 2x 3. j 20 4j

4.

5d

40

5d

5. 9m

28

2m

6. 9 x

8

8

4 x 9

Solve. 7. 8k 6k 26 8. 7p 25 12p 9. 3r 18 6r

10. 8w

9

3w

16

11. 32

5v

3v

8

12.

12y

10

6y

14

13. 8 a

5

6

3 a 4

14. 4 n

1

10

2 n 3

15. 20

1 d 5

3 d 10

16

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28

Holt Middle School Math Course 2

Name

LESSON

Date

Class

Reteach

y

rise 4 2 4 2 2 right (4, 4)

run

12-5 Slope

The slope of a line is a ratio that measures the steepness of that line. The sign of the slope tells whether the line is rising or falling. You can find the slope of a line by comparing any two points on that line. Find the slope of the line between (2, 1) and (4, 4). slope

rise run up ( ) or down ( ) right ( ) or left ( )

4 4 1 2 3 2

3 up 2

(2, 1) 4

x

O

2 4

Use the graph to complete the statements. 1. You have to move up slope

rise run

and right

to go from one point to the other.

3 2

up ( ) or down ( ) right ( ) or left ( )

2. The slope of the line is

.

You can find the y-intercept of a line by locating the point where it crosses the y-axis. The y-intercept is the y-coordinate of that point. 3. The line crosses the y-axis at the point ( ). The y-intercept is .

Tell the slope and y-intercept of each line given its graph. 4.

4 2 (0, 1) 4 2

y

5.

4 2

y

x

2 (1, 4 2) 4 2 ( 2,

x O

2) 4 (2, 2 2) 4

O

2 4

slope

up or down right or left

slope

up or down right or left

y-intercept

Copyright © by Holt, Rinehart and Winston. All rights reserved.

y-intercept 48

Holt Middle School Math Course 2

Name

LESSON

Date

Class

Reteach

12-5 Slope (continued)

You can find the slope and y-intercept of a line from the equation when the equation is in the form y mx b. This is called slope-intercept form. The variable m represents the slope, and b represents the y-intercept. y y mx 4x b 2

2

y y

mx

2 x 3

b

The slope is 4, and the y-intercept is 2.

The slope is 3 , and the y-intercept is 0.

Tell the slope and the y-intercept of each line, given the equation. 6. y 5x 4 7. y

1 x 2

1

8. y

3 x 2

Using the slope and the y-intercept, you can graph the equation y

3 x 4

y

3.

4 2 4 2 3 up 2 4 4 right (4, 0) x O 2 4 (0, 3)

The y-intercept is 3, so the graph contains the point (0, 3). The slope is 4 . Starting at (0, 3), count up 3, then 4 to the right. This is the point (4, 0). Then draw a line through the two points.

3

9. Graph y 3x and y-intercept. y-intercept slope

5 using the slope

y

; plot point (

1

)

4 2

From the y-intercept, count down and right . Plot that point and draw

x

4 2

O

2

2

4

a line through the two points.

Copyright © by Holt, Rinehart and Winston. All rights reserved.

49

Holt Middle School Math Course 2

Name

LESSON

Date

Class

Practice B

12-5 Slope

Tell the slope and y-intercept of each line, given its graph. 1.

4 2 4 2

y

(1, 3) (2, 2) (3, 1)

2.

4 2

y

(3, 2)

x O

2 4 2 4 4 2

x O

2 (0, 4 2 2) 4

Tell the slope and y-intercept, given the equation. 3. y 2x 3 4. y

1 x 3

5

5. y

4x

6. y

4x

1

7. y

x

4

8. y

2x

8

Graph each line, given the equation. 9. y

1 x 2

5

y

10. y

4

y

4

4 2 2

x x

4 2

O

2 4

2

4

4

2

O

2

2

4

Copyright © by Holt, Rinehart and Winston. All rights reserved.

46

Holt Middle School Math Course 2

Name

LESSON

Date

Class

Reteach

The graph of a nonlinear function is not a straight line. The slope of a nonlinear function changes as the line curves.

y

6 4

12-6 Nonlinear Functions

The graph of a linear function is a straight line. The slope of the line is constant.

y

4 2

x

4 2

O

2 4

2

4 4 2

2

x O

2 2 4

Tell whether the graph is linear or nonlinear. 1.

6 4 2 4 2

y

2.

4 2

y

x O

2 4

x

4 2

2

4

O

2

2

4

Is the graph a straight line? The graph is .

Is the graph a straight line? The graph is .

Copyright © by Holt, Rinehart and Winston. All rights reserved.

59

Holt Middle School Math Course 2

Name

LESSON

Date

Class

Practice A

12-6 Nonlinear Functions

Tell whether the graph is linear or nonlinear. 1. This is the graph of y

y

4 2 4 2

3x

3.

2. This is the graph of y

y

x 2.

x

4 2

x

4 2

O

2 4

2

4

O

2 4

2

4

Is the graph a straight line?

Is the graph a straight line?

3. This is the graph of y

y

4 2

x2

1.

4. This is the graph of y

y

4 2

x.

x

4 2

x

4 2

O

2 4

2

4

O

2 4

2

4

Is the graph a straight line?

Is the graph a straight line?

Tell whether the function in each table could be linear or is nonlinear. 5. x 5 6 7 y 25 36 49 6. x 2 4 6 y 100 300 500

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56

Holt Middle School Math Course 2

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