Warm Up

Evaluating Algebraic Expressions

1-9 Solving Two-Step Equations

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California StandardsCalifornia Standards

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Warm UpAdd or subtract.

1. –6 + (–5) 2. 4 – (–3)3. –2 + 11

Multiply or divide.4. –5(–4)

5.

6. 7(–8)

–1179

20

–6

–56

18–3



AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results.Also covered: AF1.1

California Standards



Two-step equations contain two operations.

For example, the equation 6x 2 = 10 contains multiplication and subtraction.

6x 2 = 10

Subtraction

Multiplication



Translate the sentence into an equation.

17 less than the quotient of a number x and 2 is 21.

Additional Example 1A: Translating Sentences into Two-Step Equations


(x ÷ 2) – 17 = 21

x 2

17 = 21




Twice a number m increased by –4 is 0.

Additional Example 1B: Translating Sentences into Two-Step Equations

Twice a number m increased by –4 is 0.

2 ● m + (–4) = 0

2m + (–4) = 0




7 more than the product of 3 and a number t is 21.

Check It Out! Example 1A

7 more than the product of 3 and a number t is 16.

3 ● t + 7 = 16

3t + 7 = 16





Check It Out! Example 1B


(x ÷ 4) – 3 = 7

x 4

3 = 7



Solve 3x + 4 = –11.

Additional Example 2A: Solving Two-Step Equations Using Division

3x + 4 = –11Step 1: Note that x is multiplied by 3. Then 4 is added. Work backward: Since 4 is added to 3x, subtract 4 from both sides.

– 4 – 4

3x = –15

Step 2: 3x = –153 3

x = –5

Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.



Solve 8 = –5y – 2.

Additional Example 2B: Solving Two-Step Equations Using Division

8 = –5y – 2Since 2 is subtracted from –5y, add 2 to both sides to undo the subtraction. + 2 + 2

10 = –5y

10 = –5y

–5 –5

–2 = y or

Since y is multiplied by –5, divide both sides by –5 to undo the multiplication.

y = –2



Solve 7x + 1 = –13.


7x + 1 = –13Step 1: Note that x is multiplied by 7. Then 1 is added. Work backward: Since 1 is added to 7x, subtract 1 from both sides.

– 1 – 1

7x = –14

Step 2: 7x = –147 7

x = –2

Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.



Solve 12 = –5y – 3.


12 = –5y – 3 Since 3 is subtracted from –5y, add 3 to both sides to undo the subtraction.

+ 3 + 3

15 = –5y

15 = –5y–5 –5

–3 = y or

Since y is multiplied by –5, divide both sides by –5 to undo the multiplication.

y = –3



Solve 4 + = 9.

Additional Example 3A: Solving Two-Step Equations Using Multiplication

Step 1:

– 4 – 4

Step 2:

m = 35

Since m is divided by 7, multiply both sides by 7 to undo the division.

m 7

4 + = 9m7

= 5m7

Note that m is divided by 7. Then 4 is added. Work backward: Since 4 is added to , subtract 4 from both

sides.

m7

(7) = 5(7)m7



Solve 14 = – 3.

Additional Example 3B: Solving Two-Step Equations Using Multiplication

Step 1:

+ 3 + 3

Step 2:

34 = z

z is divided by 2, multiply both sides by 2 to undo the division.

z 2

14 = – 3 z 12

17 = z 2

Since 3 is subtracted from t , add 3 to both sides to

undo the subtraction.

z2

(2)17 = (2)z 2



Solve 2 + = 9.


Step 1:

– 2 – 2

Step 2:

k = 42

Since k is divided by 6, multiply both sides by 6 to undo the division.

k 6

2 + = 9k 6

= 7k 6

Note that k is divided by 6. Then 2 is added. Work backward. Since 2 is added to , subtract 2 from both

sides.

k 6

(6) = 7(6)k 6



Solve 10 = – 2.


Step 1:

+ 2 + 2

Step 2:

48 = p

p is divided by 4, multiply both sides by 4 to undo the division.

p 4

10 = – 2 p 14

12 = p 4

Since 2 is subtracted from t , add 2 to both sides to

undo the subtraction.

p4

(4)12 = (4)p 4



Donna buys a portable DVD player that costs $120. She also buys several DVDs that cost $14 each. She spends a total of $204. How many DVDs does she buy?

Additional Example 4: Consumer Math Application

Let d represent the number of DVDs that Donna buys. That means Donna can spend $14d plus the cost of the DVD player.

cost of DVD player

cost of DVDs

total cost+ =

$120 14d $204+ =



Donna buys a portable DVD player that costs $120. She also buys several DVDs that cost $14 each. She spends a total of $204. How many DVDs does she buy?

Additional Example 4 Continued

$120 14d $204+ =

120 + 14d = 204

–120 –120

14d = 8414d = 8414 14

d = 6 Donna purchased 6 DVDs.



John buys an MP3 player that costs $249. He also buys several songs that cost $0.99 each. He spends a total of $277.71. How many songs does he buy?

Check It Out! Example 4

Let s represent the number of songs that John buys. That means John can spend $0.99s plus the cost of the MP3 player.

cost of MP3 player

cost of songs

total cost+ =

$249 0.99s $277.71+ =



249 + 0.99s = 277.71

–249 –249

0.99s = 28.71

s = 29 John purchased 29 songs.

John buys an MP3 player that costs $249. He also buys several songs that cost $0.99 each. He spends a total of $277.71. How many songs does he buy?

Check It Out! Example 4 Continued

$249 0.99s $277.71+ =

0.99s = 28.710.99 0.99



Lesson QuizTranslate the sentence into an equation.

1. The product of –3 and a number c, plus 14, is –7.

Solve.

2. 17 = 2x – 3 3. –4m + 3 = 15

4. – 5 = 1 5. 2 = 3 –

–3c + 14 = –7

12 4

10 –3w2

x4

6. A discount movie pass costs $14. With the

pass, movie tickets cost $6 each. Fern spent a

total of $68 on the pass and movie tickets.

How many movies did he see? 9

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