Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers AF4.1 Solve two-step linear equations and inequalities in one variable over

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

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.

California Standards



Recall that two-step equations contain two operations, and therefore, require two inverse operations to solve. Before solving, ask yourself, “What is being done to the variable, and in what order?” One method to solve the equation is to work backward to undo the operations.



The mechanic’s bill to repair Mr. Wong’s car was $653.05. The mechanic charges $45.50 an hour for labor, and the parts that were used cost $443.75. How many hours did the mechanic work on the car?

Additional Example 1: Problem Solving Application



Additional Example 1 Continued

11 Understand the Problem

The answer is the number of hours the mechanic worked on the car.

List the important information:

Let h represent the hours the mechanic worked.

• The parts cost $443.75.• The labor cost $45.50 per hour.• The total bill was $653.05.

Total bill = Parts + Labor653.05 = 443.75 + 45.50h



Think: First the variable is multiplied by 45.50, and then 443.75 is added to the result. Work backward to solve the equation. Undo the operations in reverse order: First subtract 443.75 from both sides of the equation, and then divide both sides of the new equation by 45.50.

22 Make a Plan




653.05 = 443.75 + 45.50h

Solve33

–443.75 –443.75 209.30 = 45.50h

4.6 = h

The mechanic worked for 4.6 hours on Mr. Wong’s car.


Since h is multiplied by 45.50, divide both sides by 45.50.

209.30 45.50h45.50 45.50

=

Since 443.75 is added to both sides, subtract 443.75 from both sides.



You can use a table to decide whether your answer is reasonable.

Look Back44


Hours Labor Parts Total Cost

1 $45.50 $443.75 $489.25

2 $91.00 $443.75 $534.75

3 $136.50 $443.75 $580.25

4 $182.00 $443.75 $625.75

5 $227.50 $443.75 $671.25

4.6 hours is a reasonable answer.



The mechanic’s bill to repair your car was $850. The mechanic charges $35 an hour for labor, and the parts that were used cost $275. How many hours did the mechanic work on your car?

Check It Out! Example 1



Check It Out! Example 1 Continued

11 Understand the Problem

The answer is the number of hours the mechanic worked on your car.

List the important information:

Let h represent the hours the mechanic worked.

• The parts cost $275.

• The labor cost $35 per hour.

• The total bill was $850.

Total bill = Parts + Labor

850 = 275 + 35h



Think: First the variable is multiplied by 35, and then 275 is added to the result. Work backward to solve the equation. Undo the operations in reverse order: First subtract 275 from both sides of the equation, and then divide both sides of the new equation by 35.

22 Make a Plan




850 = 275 + 35h

Solve33

–275 –275575 = 35h

16.4 h

The mechanic worked for about 16.4 hours on your car.


575 35h35 35=

Since h is multiplied by 35, divide both sides by 35.

Since 275 is added to both sides, subtract 275 from both sides.



Look Back44


You can use a table to decide whether your answer is reasonable.

Hours Labor Parts Total Cost

13 $455 $275 $730

14 $490 $275 $765

15 $525 $275 $800

16 $560 $275 $835

17 $595 $275 $870

16.4 hours is a reasonable answer.



Additional Example 2A: Solving Two-Step Equations

Solve + 7 = 22.n3

+ 7 – 7 = 22 – 7n3

Since n is divided by 3, multiply both sides by 3.3 = 3 15

n3n = 45

Method 1: Use fraction operations.

+ 7 = 22n3

Since 7 is added to ,

subtract 7 from both sides to undo the addition.

n3

= 15n3



Additional Example 2B: Solving Two-Step Equations

Solve = 9.y – 4 3

= 9y – 43

y – 4 = 27

+ 4 + 4 Since 4 is subtracted from y, add 4 to both sides to undo

the subtraction.y = 31

Multiply both sides by the denominator.

Method 2: Multiply both sides of the equation by the denominator.

= 9y – 43(3) (3)



Check It Out! Example 2A

Solve + 8 = 18.n4

+ 8 – 8 = 18 – 8n4

Since n is divided by 4, multiply both sides by 4.4 = 4 10

n4

n = 40

Method 1: Use fraction operations.

+ 8 = 18n4

Since 8 is added to ,

subtract 8 from both sides to undo the addition.

n4

= 10n4



Check It Out! Example 2B

Solve = 7.y – 7 2

= 7y – 72

y – 7 = 14

+ 7 + 7 Since 7 is subtracted from y, add 7 to both sides to undo the subtraction.y = 21

Multiply both sides by the denominator.

Method 2: Multiply both sides of the equation by the denominator.

= 7y – 72(2) (2)

Documents

Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers AF4.1 Solve two-step linear equations and inequalities in one variable over