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This pdf includes the following topics:-

Identifying Inverse Operations

Solving Two-Step Equations

Analyzing a Video Game

Solving Two-Step Equations

Solving Equations by Combining Like Terms

Identifying Inverse Operations

Solving Two-Step Equations

Analyzing a Video Game

Solving Two-Step Equations

Solving Equations by Combining Like Terms

1.
English Spanish

7.4 Solving Two-Step Equations

What is a “two-step”

S

STATE equation? How can you solve a two-step equation?

STANDARDS

MA.6.A.3.2 Sir Isaac Newton’s Third Law of Motion

For every action, there is an equal and opposite reaction.

A teddy bear Chair says “I’m ready”.

Sits in a chair. With a confident “Yup”

Down pushes Teddy. The chair pushes up.

5 lb

5 lb Sir Isaac Newton

(1642–1727)

Because 5 − 5 = 0, neither the bear nor the chair moves.

1 ACTIVITY: Identifying Inverse Operations

Work with a partner. Describe how you can “undo” the operation in blue.

a. Sample: 3x + 5 =14 3x + 5 = 14

Subtract 5 from each side. −5 −5

3x = 9

x−2

b. 2n − 6 = 4 c. 2(m + 3) = 6 d. — = 1

4

2 ACTIVITY: Solving Two-Step Equations

Work with a partner. Solve each equation in Activity 1. Use substitution to

check your answer.

a. 3x + 5 = 14 b. 2n − 6 = 4

c. 2(m + 3) = 6 d. (x − 2) ÷ 4 = 1

296 Chapter 7 Equations

7.4 Solving Two-Step Equations

What is a “two-step”

S

STATE equation? How can you solve a two-step equation?

STANDARDS

MA.6.A.3.2 Sir Isaac Newton’s Third Law of Motion

For every action, there is an equal and opposite reaction.

A teddy bear Chair says “I’m ready”.

Sits in a chair. With a confident “Yup”

Down pushes Teddy. The chair pushes up.

5 lb

5 lb Sir Isaac Newton

(1642–1727)

Because 5 − 5 = 0, neither the bear nor the chair moves.

1 ACTIVITY: Identifying Inverse Operations

Work with a partner. Describe how you can “undo” the operation in blue.

a. Sample: 3x + 5 =14 3x + 5 = 14

Subtract 5 from each side. −5 −5

3x = 9

x−2

b. 2n − 6 = 4 c. 2(m + 3) = 6 d. — = 1

4

2 ACTIVITY: Solving Two-Step Equations

Work with a partner. Solve each equation in Activity 1. Use substitution to

check your answer.

a. 3x + 5 = 14 b. 2n − 6 = 4

c. 2(m + 3) = 6 d. (x − 2) ÷ 4 = 1

296 Chapter 7 Equations

2.
English Spanish

3 ACTIVITY: Analyzing a Video Game

Work with a partner. For Level 1 in a video game, you have to accomplish a

sequence of challenges. Then, you have to leave the level by undoing the

challenges in reverse order.

a. Describe the challenges in order.

b. Describe the order of challenges to get out of the level.

2. Pick up

the map. 3. Put on

1. Take the the shield.

blue crystal.

c. This is Level 1. Make up challenges for Level 2. Draw the level and describe

the reverse order to get back out of the level.

4. IN YOUR OWN WORDS What is a “two-step” equation? How can you solve a

two-step equation? Give an example to show how your procedure works.

“Hey, it says ‘Close this flap first,’

but they closed it last!”

Use what you learned about solving two-step equations to complete

Exercises 5 –7 on page 301.

Section 7.4 Solving Two-Step Equations 297

3 ACTIVITY: Analyzing a Video Game

Work with a partner. For Level 1 in a video game, you have to accomplish a

sequence of challenges. Then, you have to leave the level by undoing the

challenges in reverse order.

a. Describe the challenges in order.

b. Describe the order of challenges to get out of the level.

2. Pick up

the map. 3. Put on

1. Take the the shield.

blue crystal.

c. This is Level 1. Make up challenges for Level 2. Draw the level and describe

the reverse order to get back out of the level.

4. IN YOUR OWN WORDS What is a “two-step” equation? How can you solve a

two-step equation? Give an example to show how your procedure works.

“Hey, it says ‘Close this flap first,’

but they closed it last!”

Use what you learned about solving two-step equations to complete

Exercises 5 –7 on page 301.

Section 7.4 Solving Two-Step Equations 297

3.
English Spanish

7.4 Lesson

Lesson Tutorials

Key Vocabulary

two-step equation,

p. 298 Solving Two-Step Equations

terms, p. 300 A two-step equation is an equation that contains two different

like terms, p. 300 operations. To solve a two-step equation, use inverse operations

to isolate the variable.

EXAMPLE 1 Solving Two-Step Equations

a. Solve 2x − 5 = 13.

2x − 5 = 13 Write the equation.

Step 1: Undo the subtraction. +5 +5 Add 5 to each side.

Check

2x = 18 Simplify. 2x − 5 = 13

2x 18 ?

Step 2: Undo the multiplication. — =— Divide each side by 2. 2(9) − 5 = 13

2 2

?

x= 9 Simplify. 18 − 5 = 13

The solution is x = 9.

13 = 13 ✓

y

Common Error b. Solve 4 = — + 1.

8

Do not get confused

y

when the variable is 4=—+1 Write the equation.

on the right side of the 8

equation. The equation −1 −1 Subtract 1 from each side.

y Check

4 = — + 1 is solved the y

8

3=—

y

Simplify. 4=—+1

same way as the 8

y 8

equation — + 1 = 4. ? 24

8 4=—+1

⋅

3 8=— 8

y

8 ⋅ Multiply each side by 8.

?

8

4=3+1

24 = y Simplify.

4=4 ✓

The solution is y = 24.

Solve the equation. Check your solution.

Exercises 5–16 h

1. 5c − 1 = 14 2. — + 9 = 20 3. 3(x − 1) = 9

4

298 Chapter 7 Equations

7.4 Lesson

Lesson Tutorials

Key Vocabulary

two-step equation,

p. 298 Solving Two-Step Equations

terms, p. 300 A two-step equation is an equation that contains two different

like terms, p. 300 operations. To solve a two-step equation, use inverse operations

to isolate the variable.

EXAMPLE 1 Solving Two-Step Equations

a. Solve 2x − 5 = 13.

2x − 5 = 13 Write the equation.

Step 1: Undo the subtraction. +5 +5 Add 5 to each side.

Check

2x = 18 Simplify. 2x − 5 = 13

2x 18 ?

Step 2: Undo the multiplication. — =— Divide each side by 2. 2(9) − 5 = 13

2 2

?

x= 9 Simplify. 18 − 5 = 13

The solution is x = 9.

13 = 13 ✓

y

Common Error b. Solve 4 = — + 1.

8

Do not get confused

y

when the variable is 4=—+1 Write the equation.

on the right side of the 8

equation. The equation −1 −1 Subtract 1 from each side.

y Check

4 = — + 1 is solved the y

8

3=—

y

Simplify. 4=—+1

same way as the 8

y 8

equation — + 1 = 4. ? 24

8 4=—+1

⋅

3 8=— 8

y

8 ⋅ Multiply each side by 8.

?

8

4=3+1

24 = y Simplify.

4=4 ✓

The solution is y = 24.

Solve the equation. Check your solution.

Exercises 5–16 h

1. 5c − 1 = 14 2. — + 9 = 20 3. 3(x − 1) = 9

4

298 Chapter 7 Equations

4.
English Spanish

EXAMPLE 2 Standardized Test Practice

You pay $80 for a game system. The monthly rental fee for games

is m dollars. Your cost for the year is $188. Using the equation

12m + 80 = 188, how much is your monthly fee?

A $8

○ B $9

○ C $12

○ D $22

○

12m + 80 = 188 Write the equation.

− 80 − 80 Subtract 80 from each side.

12m = 108 Simplify.

12m 108

—=— Divide each side by 12.

12 12

m=9 Simplify.

Your monthly fee is $9. The correct answer is ○

B .

EXAMPLE 3 Real-Life Application

You and your friend rent a tandem bike. Your total cost is $42.

Write and solve an equation to find the number of extra hours you

rented the bike.

Words The cost plus the cost times the number is the

for three for each of extra total

hours extra hour hours cost.

Variable Let h be the number of extra hours.

$24 for 3 hours

$4.50 each extra hour

Equation 24 + 4.5 ⋅ h = 42

24 + 4.5h = 42 Write the equation.

− 24 − 24 Subtract 24 from each side.

4.5h = 18 Simplify.

4.5h 18

—=— Divide each side by 4.5.

4.5 4.5

h=4 Simplify.

You rented the bike for 4 extra hours.

4. You and your friend rent a kayak. It costs $40 for the first 4 hours

Exercises 19 and $7.50 for each extra hour. Your total cost is $62.50. Write and

and 20 solve an equation to find the number of extra hours you rented

the kayak.

Section 7.4 Solving Two-Step Equations 299

EXAMPLE 2 Standardized Test Practice

You pay $80 for a game system. The monthly rental fee for games

is m dollars. Your cost for the year is $188. Using the equation

12m + 80 = 188, how much is your monthly fee?

A $8

○ B $9

○ C $12

○ D $22

○

12m + 80 = 188 Write the equation.

− 80 − 80 Subtract 80 from each side.

12m = 108 Simplify.

12m 108

—=— Divide each side by 12.

12 12

m=9 Simplify.

Your monthly fee is $9. The correct answer is ○

B .

EXAMPLE 3 Real-Life Application

You and your friend rent a tandem bike. Your total cost is $42.

Write and solve an equation to find the number of extra hours you

rented the bike.

Words The cost plus the cost times the number is the

for three for each of extra total

hours extra hour hours cost.

Variable Let h be the number of extra hours.

$24 for 3 hours

$4.50 each extra hour

Equation 24 + 4.5 ⋅ h = 42

24 + 4.5h = 42 Write the equation.

− 24 − 24 Subtract 24 from each side.

4.5h = 18 Simplify.

4.5h 18

—=— Divide each side by 4.5.

4.5 4.5

h=4 Simplify.

You rented the bike for 4 extra hours.

4. You and your friend rent a kayak. It costs $40 for the first 4 hours

Exercises 19 and $7.50 for each extra hour. Your total cost is $62.50. Write and

and 20 solve an equation to find the number of extra hours you rented

the kayak.

Section 7.4 Solving Two-Step Equations 299

5.
English Spanish

Terms and Like Terms

In the equation 5x + 2x = 16 − 2, 5x, 2x, 16, and 2 are called terms.

5x and 2x are called like terms. 16 and 2 are also like terms.

Like terms Like terms

5x + 2x = 16 − 2

To solve, use the Distributive Property to combine like terms.

EXAMPLE 4 Solving Equations by Combining Like Terms

a. Solve the equation 3x + 6x = 45.

3x + 6x = 45 Write the equation.

(3 + 6)x = 45 Use the Distributive Property to combine like terms.

9x = 45 Simplify.

Check

9x

—=—

45

Divide each side by 9. 3x + 6x = 45

9 9 ?

3(5) + 6(5) = 45

x=5 Simplify.

?

15 + 30 = 45

The solution is x = 5.

45 = 45 ✓

b. Solve the equation 5a − 2a = 6.

5a − 2a = 6 Write the equation.

(5 − 2)a = 6 Use the Distributive Property to combine like terms.

3a = 6 Simplify. Check

3a 6

—=— Divide each side by 3. 5a − 2a = 6

3 3 ?

5(2) − 2(2) = 6

a=2 Simplify.

?

10 − 4 = 6

The solution is a = 2.

6=6 ✓

Solve the equation. Check your solution.

Exercises 21–29 5. 2x + 5x = 7 6. 7c + 4c = 22 7. 3w − 2w = 9

300 Chapter 7 Equations

Terms and Like Terms

In the equation 5x + 2x = 16 − 2, 5x, 2x, 16, and 2 are called terms.

5x and 2x are called like terms. 16 and 2 are also like terms.

Like terms Like terms

5x + 2x = 16 − 2

To solve, use the Distributive Property to combine like terms.

EXAMPLE 4 Solving Equations by Combining Like Terms

a. Solve the equation 3x + 6x = 45.

3x + 6x = 45 Write the equation.

(3 + 6)x = 45 Use the Distributive Property to combine like terms.

9x = 45 Simplify.

Check

9x

—=—

45

Divide each side by 9. 3x + 6x = 45

9 9 ?

3(5) + 6(5) = 45

x=5 Simplify.

?

15 + 30 = 45

The solution is x = 5.

45 = 45 ✓

b. Solve the equation 5a − 2a = 6.

5a − 2a = 6 Write the equation.

(5 − 2)a = 6 Use the Distributive Property to combine like terms.

3a = 6 Simplify. Check

3a 6

—=— Divide each side by 3. 5a − 2a = 6

3 3 ?

5(2) − 2(2) = 6

a=2 Simplify.

?

10 − 4 = 6

The solution is a = 2.

6=6 ✓

Solve the equation. Check your solution.

Exercises 21–29 5. 2x + 5x = 7 6. 7c + 4c = 22 7. 3w − 2w = 9

300 Chapter 7 Equations

6.
English Spanish

7.4 Exercises

Help with Homework

1. VOCABULARY Why is the equation 5x − 12 = 23 called a two-step equation?

2. VOCABULARY Identify the like terms in the equation 3x + 4x = 21. Explain

why they are like terms.

3. WHICH ONE DOESN’T BELONG? Which one does not belong with the other

three? Explain your reasoning.

16x − 5x = 22 11x = 22 11(x − 1) = 22 (16 − 5)x = 22

4. WRITING Describe a process you can use to combine the like terms in the

equation 16x − 5x = 22.

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Solve the equation. Check your solution.

z a

1 5. 8 + — = 23 6. — − 9 = 12 7. 4c − 7 = 17

4 3

x

8. 6 + — = 31 9. 4b − 12 = 0 10. 12w − 8 = 28

5

t t

11. — − 9 = 13 12. 131 = 7s + 12 13. 42 + — = 54

19 9

s

14. 2.4a + 8 = 27.2 15. — − 0.6 = 1.2 16. 5t − 17.2 = 16.3

3

ERROR ANALYSIS Describe and correct the error in solving the equation.

17. 18.

✗ ✗ 28y + 7 = 21

y

4=—+1

8 28y = 28

32 = y + 1 y=1

31 = y

2 3 19. HIKING You go on a hike with your uncle. Your

backpack weighs 25 pounds. Your uncle is a math

teacher and he tells you that your pack is 7 pounds

less than twice as heavy as his pack. Use the equation

2p − 7 = 25 to find the weight of your uncle’s backpack.

Section 7.4 Solving Two-Step Equations 301

7.4 Exercises

Help with Homework

1. VOCABULARY Why is the equation 5x − 12 = 23 called a two-step equation?

2. VOCABULARY Identify the like terms in the equation 3x + 4x = 21. Explain

why they are like terms.

3. WHICH ONE DOESN’T BELONG? Which one does not belong with the other

three? Explain your reasoning.

16x − 5x = 22 11x = 22 11(x − 1) = 22 (16 − 5)x = 22

4. WRITING Describe a process you can use to combine the like terms in the

equation 16x − 5x = 22.

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Solve the equation. Check your solution.

z a

1 5. 8 + — = 23 6. — − 9 = 12 7. 4c − 7 = 17

4 3

x

8. 6 + — = 31 9. 4b − 12 = 0 10. 12w − 8 = 28

5

t t

11. — − 9 = 13 12. 131 = 7s + 12 13. 42 + — = 54

19 9

s

14. 2.4a + 8 = 27.2 15. — − 0.6 = 1.2 16. 5t − 17.2 = 16.3

3

ERROR ANALYSIS Describe and correct the error in solving the equation.

17. 18.

✗ ✗ 28y + 7 = 21

y

4=—+1

8 28y = 28

32 = y + 1 y=1

31 = y

2 3 19. HIKING You go on a hike with your uncle. Your

backpack weighs 25 pounds. Your uncle is a math

teacher and he tells you that your pack is 7 pounds

less than twice as heavy as his pack. Use the equation

2p − 7 = 25 to find the weight of your uncle’s backpack.

Section 7.4 Solving Two-Step Equations 301

7.
English Spanish

20. HELICOPTER TOUR You pilot a helicopter tour from

Orlando to West Palm Beach along the coast. On Orlando

the return trip, you fly straight back to Orlando at

a steady speed in 1.3 hours. The total distance is

313 miles. Write and solve an equation to find 170 mi

your speed from West Palm Beach to Orlando.

West Palm Beach

Solve the equation. Check your solution.

4 21. c + 3c = 16 22. 2x + 6x = 24 23. 51 = 15y + 2y

24. 6z − 5z = 20 25. 18 = 8a − 5a 26. 7t − t = 54

27. 3.2x − 1.2x = 8 28. 4.8 = 1.8n + 0.6n 29. 15 = 3.5s − 2s

30. COMPUTERS You help the owner of a computer store load monitors into a

truck. You load 10 monitors and the owner loads 7 monitors. The total weight

of the monitors is 765 pounds. Write and solve an equation to find the weight

of each monitor.

d d

31. MODEL TRAIN The model train track has 6

straight sections and 12 curved sections. The

total length of the track is 351 centimeters.

Each section is d centimeters long. Write

and solve an equation to find the length of

each section of the track.

Solve the equation. Check your solution.

g h

32. 32y + 10 − 2 = 24 33. 11 + — − 3 = 12 34. 9.2 = 5.7 + — + 0.4

4 6

z−3

35. 125 = 5(3 + x) 36. 12(z − 7) = 60 37. — = 10

10

(5 + a) 22 + t

38. 7 = — 39. 6(11 + s) = 96 40. 15 = —

4 3

Write and solve an equation to find x.

41. Perimeter = 18 inches 42. Perimeter = 35 feet 43. Perimeter = 132 yards

x x

7 in. 3x 4x

5 in.

15 ft

4x

2x

302 Chapter 7 Equations

20. HELICOPTER TOUR You pilot a helicopter tour from

Orlando to West Palm Beach along the coast. On Orlando

the return trip, you fly straight back to Orlando at

a steady speed in 1.3 hours. The total distance is

313 miles. Write and solve an equation to find 170 mi

your speed from West Palm Beach to Orlando.

West Palm Beach

Solve the equation. Check your solution.

4 21. c + 3c = 16 22. 2x + 6x = 24 23. 51 = 15y + 2y

24. 6z − 5z = 20 25. 18 = 8a − 5a 26. 7t − t = 54

27. 3.2x − 1.2x = 8 28. 4.8 = 1.8n + 0.6n 29. 15 = 3.5s − 2s

30. COMPUTERS You help the owner of a computer store load monitors into a

truck. You load 10 monitors and the owner loads 7 monitors. The total weight

of the monitors is 765 pounds. Write and solve an equation to find the weight

of each monitor.

d d

31. MODEL TRAIN The model train track has 6

straight sections and 12 curved sections. The

total length of the track is 351 centimeters.

Each section is d centimeters long. Write

and solve an equation to find the length of

each section of the track.

Solve the equation. Check your solution.

g h

32. 32y + 10 − 2 = 24 33. 11 + — − 3 = 12 34. 9.2 = 5.7 + — + 0.4

4 6

z−3

35. 125 = 5(3 + x) 36. 12(z − 7) = 60 37. — = 10

10

(5 + a) 22 + t

38. 7 = — 39. 6(11 + s) = 96 40. 15 = —

4 3

Write and solve an equation to find x.

41. Perimeter = 18 inches 42. Perimeter = 35 feet 43. Perimeter = 132 yards

x x

7 in. 3x 4x

5 in.

15 ft

4x

2x

302 Chapter 7 Equations

8.
English Spanish

44. TRADING CARDS You have 80 trading cards. Your friend says that you have

16 less than 4 times the number of cards that she has. You say that you have

8 more than 3 times as many cards as she has. Can you both be right? Explain.

45. RECIPE You want to make 3 batches

Barbecue sauce of barbecue sauce, but you can’t

(1 batch) remember how much brown sugar

2.5 cups of catsup

? cups of brown sugar y

you need. You know that 4 batches

0.25 cup of diced onions

Add a taste of mustard make about 17 cups of sauce. How

much brown sugar do you need for

3 batches?

46. TESTS After four 100-point tests, you have 365 points.

a. How many points do you need to score on your next 100-point test

to have a mean score of 92 points?

b. Would a mean score of 92 points after 5 tests be greater than or

less than your mean score after 4 tests?

c. Your score on each test is a whole number. Is it possible that your

mean score does not change after the fifth test? Explain.

47. HARDCOVER BOOK Each page of the book has the same thickness t.

a. What other piece of information

do you need to find the thickness 3 cm

of one page? 3 mm

b. Choose a reasonable number for

the missing piece of information.

c. Use the number from part (b) to write and solve an equation to find

the thickness t of one page. Does your answer seem reasonable?

48. A teacher has a box of pens and pencils. There are 8 more

pencils than pens. After students take 1 pen and 5 pencils from the box,

there are 26 pens and pencils left in the box. How many pens are in

the box now? How many pencils?

Write the percent as a fraction or mixed number in simplest form. SECTION 4.1

49. 85% 50. 86% 51. 128% 52. 0.75%

53. MULTIPLE CHOICE Use a formula to find the area of 12 m

6m

the triangle. SECTION 1.5

16 m

2 2 2

○

A 36 m ○

B 48 m ○

C 72 m D 96 m2

○

Section 7.4 Solving Two-Step Equations 303

44. TRADING CARDS You have 80 trading cards. Your friend says that you have

16 less than 4 times the number of cards that she has. You say that you have

8 more than 3 times as many cards as she has. Can you both be right? Explain.

45. RECIPE You want to make 3 batches

Barbecue sauce of barbecue sauce, but you can’t

(1 batch) remember how much brown sugar

2.5 cups of catsup

? cups of brown sugar y

you need. You know that 4 batches

0.25 cup of diced onions

Add a taste of mustard make about 17 cups of sauce. How

much brown sugar do you need for

3 batches?

46. TESTS After four 100-point tests, you have 365 points.

a. How many points do you need to score on your next 100-point test

to have a mean score of 92 points?

b. Would a mean score of 92 points after 5 tests be greater than or

less than your mean score after 4 tests?

c. Your score on each test is a whole number. Is it possible that your

mean score does not change after the fifth test? Explain.

47. HARDCOVER BOOK Each page of the book has the same thickness t.

a. What other piece of information

do you need to find the thickness 3 cm

of one page? 3 mm

b. Choose a reasonable number for

the missing piece of information.

c. Use the number from part (b) to write and solve an equation to find

the thickness t of one page. Does your answer seem reasonable?

48. A teacher has a box of pens and pencils. There are 8 more

pencils than pens. After students take 1 pen and 5 pencils from the box,

there are 26 pens and pencils left in the box. How many pens are in

the box now? How many pencils?

Write the percent as a fraction or mixed number in simplest form. SECTION 4.1

49. 85% 50. 86% 51. 128% 52. 0.75%

53. MULTIPLE CHOICE Use a formula to find the area of 12 m

6m

the triangle. SECTION 1.5

16 m

2 2 2

○

A 36 m ○

B 48 m ○

C 72 m D 96 m2

○

Section 7.4 Solving Two-Step Equations 303