1. Preview
Warm Up
California Standards
Lesson Presentation

2. Warm Up Add or subtract. 1. –6 + (–5) 2. 4 – (–3) 3. –2 + 11 –11
Multiply or divide.
4. –5(–4) 5.
6. 7(–8)
–11 7 9 20 18 –3 –6
–56

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

4. Two-step equations contain two operations
Two-step equations contain two operations. For example, the equation 6x  2 = 10 contains multiplication and subtraction.
6x  2 = 10
Multiplication
Subtraction

5. Additional Example 1A: Translating Sentences into Two-Step Equations
Translate the sentence into an equation.
17 less than the quotient of a number x and 2 is 21.
17 less than the quotient of a number x and 2 is 21.
(x ÷ 2) – 17 = 21
x 2
- 17 = 21

6. Additional Example 1B: Translating Sentences into Two-Step Equations
Translate the sentence into an equation.
Twice a number m increased by –4 is 0.
Twice a number m increased by –4 is 0.
2 ● m + (–4) = 0
2m + (–4) = 0

7. Check It Out! Example 1A
Translate the sentence into an equation.
7 more than the product of 3 and a number t is 21.
7 more than the product of 3 and a number t is 16.
3 ● t + 7 = 16
3t + 7 = 16

8. Check It Out! Example 1B
Translate the sentence into an equation.
3 less than the quotient of a number x and 4 is 7.
3 less than the quotient of a number x and 4 is 7.
(x ÷ 4) – 3 = 7
x 4
- 3 = 7

9. Additional Example 2A: Solving Two-Step Equations Using Division
Solve 3x + 4 = –11.
Step 1:
3x + 4 = –11
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
3x = –15
Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.
Step 2: 3 3
x = –5

10. Additional Example 2B: Solving Two-Step Equations Using Division
Solve 8 = –5y – 2.
Since 2 is subtracted from –5y, add 2 to both sides to undo the subtraction.
8 = –5y – 2
+ 2
+ 2
10 = –5y
10 = –5y
Since y is multiplied by –5, divide both sides by –5 to undo the multiplication. –5 –5
–2 = y or
y = –2

11. Check It Out! Example 2A
Solve 7x + 1 = –13.
Step 1:
7x + 1 = –13
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
7x = –14
Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.
Step 2: 7 7
x = –2

12. Check It Out! Example 2B
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
Since y is multiplied by –5, divide both sides by –5 to undo the multiplication. –5 –5
–3 = y or
y = –3

13. Additional Example 3A: Solving Two-Step Equations Using Multiplication
Solve = 9.
= 9 m7
Step 1:
Note that m is divided by 7. Then 4 is added. Work backward: Since 4 is added to , subtract 4 from both
sides. m7
– 4
– 4
= 5 m7
(7) = 5(7) m7
Since m is divided by 7, multiply both sides by 7 to undo the division.
Step 2:
m = 35

14. Additional Example 3B: Solving Two-Step Equations Using Multiplication
z 2
Solve 14 = – 3.
14 = – 3 z
Step 1:
Since 3 is subtracted from t , add 3 to both sides to
undo the subtraction. z2
+ 3
+ 3
17 =
z 2
(2)17 = (2)
z 2
z is divided by 2, multiply both sides by 2 to undo the division.
Step 2:
34 = z

15. Check It Out! Example 3A
k 6
Solve = 9.
= 9
k 6
Step 1:
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
– 2
– 2
= 7
k 6
(6) = 7(6)
k 6
Since k is divided by 6, multiply both sides by 6 to undo the division.
Step 2:
k = 42

16. Check It Out! Example 3B
p 4
Solve 10 = – 2.
10 = – 2 p
Step 1:
Since 2 is subtracted from t , add 2 to both sides to
undo the subtraction. p4
+ 2
+ 2
12 =
p 4
(4)12 = (4)
p 4
p is divided by 4, multiply both sides by 4 to undo the division.
Step 2:
48 = p

17. Additional Example 4: Consumer Math Application
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?
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 + =

18. Additional Example 4 Continued
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?
$120
14d
$204 + =
d = 204
–120
14d = 84
14d = 84 14
d = 6
Donna purchased 6 DVDs.

19. Check It Out! Example 4
John buys an MP3 player that costs $249. He also buys several songs that cost $0.99 each. He spends a total of $ How many songs does he buy?
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 + =

20. Check It Out! Example 4 Continued
John buys an MP3 player that costs $249. He also buys several songs that cost $0.99 each. He spends a total of $ How many songs does he buy?
$249
0.99s
$277.71 + =
s =
–249
0.99s = 28.71
0.99s = 28.71
0.99
s = 29
John purchased 29 songs.

21. Lesson Quiz
Translate the sentence into an equation.
1. The product of –3 and a number c, plus 14, is –7.
Solve.
2. 17 = 2x – –4m + 3 = 15
– 5 = = 3 –
–3c + 14 = –7 10 –3 w 2 x 4 12 4
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