Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

Warm UpSolve.

1. x + 12 = 352. 8x = 1203. = 74. –34 = y + 56

x = 23x = 15y = 63y = –90

y9

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

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.

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational NumbersAdditional 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 15n

3n = 45

Use fraction operations. + 7 = 22n

3Since 7 is added to ,

subtract 7 from both sides to undo the addition.

n3

= 15n3

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational NumbersAdditional 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.

Multiply both sides of the equation by the denominator.

= 9y – 43(3) (3)

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational NumbersPartner Share! Example 2A

Solve + 8 = 18.n4

+ 8 – 8 = 18 – 8n4

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

4n = 40

Use fraction operations. + 8 = 18n

4Since 8 is added to ,

subtract 8 from both sides to undo the addition.

n4

= 10n4

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational NumbersPartner Share! 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.

Multiply both sides of the equation by the denominator.

= 7y – 72(2) (2)

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

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

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational NumbersAdditional 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

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

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

Additional Example 1 Continued

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

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.

Additional Example 1 Continued

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.

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

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

Look Back44

Additional Example 1 Continued

Hours Labor Parts Total Cost1 $45.50 $443.75 $489.252 $91.00 $443.75 $534.753 $136.50 $443.75 $580.254 $182.00 $443.75 $625.755 $227.50 $443.75 $671.25

4.6 hours is a reasonable answer.

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

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?

Partner Share! Example 1

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational NumbersPartner Share! 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 + Labor850 = 275 + 35h

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

850 = 275 + 35h

Solve33

–275 –275575 = 35h

16.4 hThe mechanic worked for about 16.4 hours on your car.

Partner Share! Example 1 Continued

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.

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

Look Back44

Partner Share! Example 1 Continued

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

Hours Labor Parts Total Cost13 $455 $275 $73014 $490 $275 $76515 $525 $275 $80016 $560 $275 $83517 $595 $275 $870

16.4 hours is a reasonable answer.

Evaluating Algebraic Expressions

2-8 Two-Step Equations with Rational Numbers

Solve.

1. – x – 3 = 10

2. 7y + 25.68 = –26.12

3. –8.3 = –3.5x + 13.4

4. = 3.1

5. The cost for a new cell phone plan is $39 per month plus a one-time start-up fee of $78. If you are charged $1032.96, how many months will the contract last?

Lesson Review!

y = –7.4x = –117

x = 6.2y = 29.1

24 months

19

y + 5 11