Course 3

2-8 Solving Two-Step Equations2-8 Solving Two-Step Equations

Course 3

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Course 3

2-8 Solving Two-Step Equations

Warm UpSolve.

1. x + 12 = 35

2. 8x = 120

3. = 7

4. –34 = y + 56

x = 23

x = 15

y = 63

y = –90

y9

Course 3

2-8 Solving Two-Step Equations

Problem of the Day

x is an odd integer. If you triple x and then subtract 7, you get a prime number. What is the smallest possible x? (Hint: What is the smallest prime number?)x = 3

Course 3

2-8 Solving Two-Step Equations

Learn to solve two-step equations.

Course 3

2-8 Solving Two-Step Equations

Sometimes more than one inverse operation is needed to solve an equation. Before solving, ask yourself, “What is being done to the variable, and in what order?” Then work backward to undo the operations.

Course 3

2-8 Solving Two-Step Equations

The mechanic’s bill to repair Mr. Wong’s car was $650. The mechanic charges $45 an hour for labor, and the parts that were used cost $443. How many hours did the mechanic work on the car?

Additional Example 1: Problem Solving Application

Course 3

2-8 Solving Two-Step Equations

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.• The labor cost $45 per hour.• The total bill was $650.

Total bill = Parts + Labor

650 = 443 + 45h

Course 3

2-8 Solving Two-Step Equations

Think: First the variable is multiplied by 45, and then 443 is added to the result. Work backward to solve the equation. Undo the operations in reverse order: First subtract 443 from both sides of the equation, and then divide both sides of the new equation by 45.

22 Make a Plan

Additional Example 1 Continued

Course 3

2-8 Solving Two-Step Equations

650 = 443 + 45h

Solve33

–443 –443 Subtract to undo the addition.

207 = 45h

4.6 = h

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

Additional Example 1 Continued

Divide to undo multiplication.207 45h45 45=

Course 3

2-8 Solving Two-Step Equations

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

Look Back44

Additional Example 1 Continued

Hours Labor Parts Total Cost

1 45 $443 $488

2 90 $443 $533

3 135 $443 $578

4 180 $443 $623

5 225 $443 $668

4.6 hours is a reasonable answer.

Course 3

2-8 Solving Two-Step Equations

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

Course 3

2-8 Solving Two-Step Equations

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

Course 3

2-8 Solving Two-Step Equations

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

Check It Out: Example 1 Continued

Course 3

2-8 Solving Two-Step Equations

850 = 275 + 35h

Solve33

–275 –275 Subtract to undo the addition.

575 = 35h

16.4 h

The mechanic worked for about 16.4 hours on your car.

Check It Out: Example 1 Continued

Divide to undo multiplication.575 35h35 35=

Course 3

2-8 Solving Two-Step Equations

Look Back44

Check It Out: Example 1 Continued

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.

Course 3

2-8 Solving Two-Step Equations

Additional Example 2A: Solving Two-Step Equations

Solve + 7 = 22

Think: First the variable is divided by 3, and then 7 is added. To isolate the variable, subtract 7, and then multiply by 3.

Subtract 7 from both sides.

n3

+ 7 – 7 = 22 – 7n3

Multiply both sides by 3.3 = 3 15n3

n = 45

Method 1: Work backward to isolate the variable.

Course 3

2-8 Solving Two-Step Equations

Additional Example 2A Continued

Multiply both sides by the denominator.

+ 7 = 22(3)n3

Subtract to undo addition.

n + 21 = 66

n = 45

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

Solve + 7 = 22n3

(3)

–21 –21

Course 3

2-8 Solving Two-Step Equations

Additional Example 2B: Solving Two-Step Equations

Solve = 9y – 4 3

Method 1: Work backward to isolate the variable.

– = 9y3

43

Rewrite the expression as the sum of two fractions.

Think: First the variable is divided by 3, and then is

subtracted. To isolate the variable, add and then multiply by 3.

43

43

43

– + = 9 +y3

43

43

(3) = (3) y3

31 t3

Add to both sides.43

Multiply both sides by 3.

y = 31

Course 3

2-8 Solving Two-Step Equations

Additional Example 2B: Solving Two-Step Equations

Solve = 9y – 4 3

= 9y – 43

y – 4 = 27

+ 4 + 4 Add to undo subtraction.

y = 31

Multiply both sides by the denominator.

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

= 9y – 43(3) (3)

Course 3

2-8 Solving Two-Step Equations

Check It Out: Example 2A

Solve + 8 = 18

Think: First the variable is divided by 4, and then 8 is added. To isolate the variable, subtract 8, and then multiply by 4.

Subtract 8 from both sides.

n4

+ 8 – 8 = 18 – 8n4

Multiply both sides by 4.4 = 4 10n4

n = 40

Method 1: Work backward to isolate the variable.

Course 3

2-8 Solving Two-Step Equations

Check It Out: Example 2A

Multiply both sides by the denominator.

+ 8 = 18(4)n4

Subtract to undo addition.

n + 32 = 72

n = 40

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

Solve + 8 = 18n4

(4)

–32 –32

Course 3

2-8 Solving Two-Step Equations

Check It Out: Example 2B

Solve = 7y – 7 2

Method 1: Work backward to isolate the variable.

– = 7y2

72

Rewrite the expression as the sum of two fractions.

Think: First the variable is divided by 2, and then is

subtracted. To isolate the variable, add and then multiply by 2.

72

72

72

– + = 7 +y2

72

72

(2) = (2) y2

21 t2

Add to both sides.72

Multiply both sides by 2.

y = 21

Course 3

2-8 Solving Two-Step Equations

Check It Out: Example 2B

Solve = 7y – 7 2

= 7y – 72

y – 7 = 14

+ 7 + 7 Add to undo subtraction.

y = 21

Multiply both sides by the denominator.

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

= 7y – 72(2) (2)

Course 3

2-8 Solving Two-Step Equations

Solve.

1. – 3 = 10

2. 7y + 25 = –24

3. –8.3 = –3.5x + 13.4

4. = 3

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 $1014, how many months will the contract last?

Lesson Quiz

y = –7

x = –117

x = 6.2

y = 28

24

x–9

y + 5 11