How do you solve a two-step equation??

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ESSENTIAL QUESTION

L E S S O N

8.2Solving Two-Step Equations

Modeling and Solving Two-Step Equations You can solve two-step equations using algebra tiles.

Use algebra tiles to model and solve 3n + 2 = 11.

Model the equation.

Remove 2 +1-tiles from

each side of the mat.

Divide each side into

3 equal groups.

The solution is n = 3.

EXAMPLEXAMPLE 1

STEP 1

STEP 2

STEP 3

STEP 4

Use algebra tiles to model and solve each equation.

YOUR TURN

1. 2x + 5 = 11 2. 3n - 1 = 8

3. 2a - 3 = -5 4. -4y + 2 = -2

Expressions, equations, and relationships—7.10.B Represent solutions for one-variable, two-step equations and inequalities on number lines. Also 7.11.A, 7.11.B

7.11.A

Since there are +1-tiles on both sides of the equation, you can remove, or subtract, 2 +1-tiles from each side to help isolate the variable.

x = 3

a = -1

n = 3

y = 1

251Lesson 8.2

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Reflect5. Analyze Relationships Describe how you could find the weight of one

baseball bat using only arithmetic. Compare your method with the one

used in Example 2.

Representing Solutions on a Number LineYou have used inverse operations to solve equations with one operation. You

can use the same method to solve equations with more than one operation.

After solving, you can represent the solution on a number line.

Tony carried 5 identical baseball bats to a ball game inside a carrying

case weighing 12 ounces. The combined weight of the bats and the case

was 162 ounces. How much did each bat weigh? Graph the solution on

a number line.

Write an equation to represent the problem.

Let w = the weight of a bat in ounces.

5 times the weight of each bat plus 12 oz is 162 oz.

5w + 12 = 162

Use inverse operations to solve the equation.

5w + 12 = 162

- 12 _____ - 12 _____

5w = 150

5w ___ 5

= 150 ____ 5

w = 30

Each bat weighed 30 ounces

Graph the solution on a number line.

EXAMPLE 2

STEP 1

STEP 2

STEP 3

Math TalkMathematical Processes

7.10.B

Could you solve the equation in Example 2 by first dividing both sides

by 5? Explain.

Subtract 12 from both sides.

Divide both sides by 5.

It is helpful to reverse the order of operations when solving equations that have more than one operation.

Unit 4252

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Write and solve an equation that represents the situation. Graph the

solution on the number line.

6. Maureen wants to buy a $198

camera. She has $30 and plans

to save $12 each week. In how

many weeks will she be able

to buy the camera?

7. A rectangular picture

frame has a perimeter of

58 inches. The height of the

frame is 18 inches. What is the

width of the frame?

YOUR TURN

Determining if a Given Value Makes an Equation True You can use substitution to decide whether a given value is the solution

of an equation.

After first doubling the weight being pulled by a dog sled, the sled driver

removes 20 pounds. The final weight of the dog sled is 180 pounds. The

equation 2w - 20 = 180 can be used to find w, the initial weight of the

sled. Determine which, if any, of these values is a solution: w = 60;

w = 80; w = 100.

Substitute each value for w in the equation 2w - 20 = 180.

w = 60 w = 80 w = 100

2(60) - 20 = 180 2(80) - 20 = 180 2(100) - 20 = 180

Evaluate to see if a true equation results.

2(60) - 20 ? =

180 2(80) - 20

? =

180 2(100) - 20

? =

180

120 - 20 ? =

180 160 - 20

? =

180 200 - 20

? =

180

100 ? =

180 140

? =

180 180

? =

180

not true not true true

The initial weight of the sled was 100 pounds.

EXAMPLEXAMPLE 3

STEP 1

STEP 2

✗✗ ✔

7.11.B

253Lesson 8.2

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Guided Practice

The equation 2x + 1 = 9 is modeled below. (Example 1)

1. To solve the equation with algebra tiles, first remove .

Then divide each side into .

2. The solution is x = .

3. 8m - 15 = 41

m =

4.

k =

k _ 3 + 21 = 27

5. 9p - 18 = 27

p = 3; p = 5; p = 7

6. a = -10; a = 0; a = 10

a ___ -2

- 5 = 0

Determine which, if any, of the given values is a solution. (Example 3)

Solve each equation. Then graph the solution on the number line. (Example 2)

Determine which, if any, of the given values is a solution.

8. 3k + 15 = 66

k = -7; k = 17; k = 27

9. p

_ 9

- 5 = 7

p = -72; p = 18; p = 108

YOUR TURN

7. How can you decide which operations to use to solve a two-step

equation?

ESSENTIAL QUESTION CHECK-IN??

Unit 4254

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Name Class Date

Independent Practice

Solve.

8. 9s + 3 = 57 9. 4d + 6 = 42 10. -3y + 12 = -48

11.

12. 13.

14. -9h - 15 = 93

15. 16.

17.

18. 19. 46 = -6t -8

k _ 2

+ 9 = 30g

__ 3

- 7 = 15 z _ 5

+ 3 = -35

24 + n __ 4

= 10 -17 + b __ 8

= 13

-5 = 9 + c _ 4

-3 + p

__ 7

= -5

8.2

27. -5.5x + 0.56 = -1.64 28. -4.2x + 31.5 = -65.1 29. k ___ 5.2

+ 81.9 = 47.2

20. After making a deposit, Puja had $264 in her savings account.

She noticed that if she added $26 to the amount originally in the

account and doubled the sum, she would get the new amount.

How much did she originally have in the account?

21. The current temperature in Smalltown is 20 °F. This is 6 degrees less

than twice the temperature that it was six hours ago. What was the

temperature in Smalltown six hours ago?

22. Daphne gave away 3 more than half of her apples. She gave away

17 apples in all. How many apples did Daphne have originally?

23. Artaud noticed that if he takes the opposite of his age and adds 40

he gets the number 28. How old is Artaud?

24. Sven has 11 more than twice as many customers as when he

started selling newspapers. He now has 73 customers. How many

did he have when he started?

25. Paula bought a ski jacket on sale for $6 less than half its original

price. She paid $88 for the jacket. What was the original price?

26. Michelle has a starting balance on a gift card for $300. She buys

several dresses at $40 a piece. After her purchases she has $140 left

on the gift card. How many dresses did she buy?

Use a calculator to solve each equation.

7.10.B, 7.11.A, 7.11.B

255Lesson 8.2

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Work Area30. Write a two-step equation involving multiplication and subtraction that

has a solution of x = 7.

31. Write a two-step equation involving division and addition that has

a solution of x = -25

32. Reason Abstractly The formula F = 1.8C + 32 allows you to find the

Fahrenheit (F) temperature for a given Celsius (C) temperature. Solve the

equation for C to produce a formula for finding the Celsius temperature

for a given Fahrenheit temperature.

33. Reason Abstractly The equation P = 2(ℓ + w) can be used to find the

perimeter P of a rectangle with length ℓ and width w. Solve the equation

for w to produce a formula for finding the width of a rectangle given its

perimeter and length.

34. Critique Reasoning A student’s solution to the equation 3x + 2 = 15

is shown. Describe the error that the student made.

35. Multiple Representations Explain how you could use the work

backward problem-solving strategy to solve the equation x _ 4 − 6 = 2.

36. Reason Abstractly Solve the equation ax + b = c for x.

FOCUS ON HIGHER ORDER THINKING

3x + 2 = 15

x + 2 = 5

x = 3

Divide both sides by 3.

Subtract 2 from both sides.

Unit 4256

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