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Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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California StandardsCalifornia Standards
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Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Two-step equations contain two operations.
For example, the equation 6x 2 = 10 contains multiplication and subtraction.
6x 2 = 10
Subtraction
Multiplication
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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
17 less than the quotient of a number x and 2 is 21.
(x ÷ 2) – 17 = 21
x 2
17 = 21
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Translate the sentence into an equation.
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Translate the sentence into an equation.
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Translate the sentence into an equation.
3 less than the quotient of a number x and 4 is 7.
Check It Out! Example 1B
3 less than the quotient of a number x and 4 is 7.
(x ÷ 4) – 3 = 7
x 4
3 = 7
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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.
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Solve 7x + 1 = –13.
Check It Out! Example 2A
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.
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Solve 12 = –5y – 3.
Check It Out! Example 2B
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Solve 2 + = 9.
Check It Out! Example 3A
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
Solve 10 = – 2.
Check It Out! Example 3B
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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+ =
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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.
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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+ =
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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
Evaluating Algebraic Expressions
1-9 Solving Two-Step Equations
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