2-8 Solving Two-Step Equations Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of...

2-8 Solving Two-Step Equations

Warm Up

Lesson Presentation

Problem of the Day

Lesson Quizzes


Warm UpSolve.

1. x + 12 = 35

2. 8x = 120

3. = 7

4. –34 = y + 56

x = 23

x = 15

y = 63

y = –90

y9


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


Learn to solve 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.


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


Additional Example 1 Continued

1 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


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.

2 Make a Plan



650 = 443 + 45h

Solve3

–443 –443 Step 1: Subtract to undo the addition.

207 = 45h

4.6 = h

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


Step 2: Divide to undo multiplication.207 45h45 45=


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

Look Back4


$668$4432255

$623$4431804

$578$4431353

$533$443902

$488$443451

Total CostPartsLaborHours

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

1 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.

2 Make a Plan



850 = 275 + 35h

Solve3

–275 –275 Step 1: Subtract to undo the addition.

575 = 35h

16.4 h

The mechanic worked for about 16.4 hours on your car.


Step 2: Divide to undo multiplication.

575 35h35 35=


Look Back4


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

$870$27559517

$835$27556016

$800$27552515

$765$27549014

$730$27545513

Total CostPartsLaborHours

16.4 hours is a reasonable answer.


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.


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 = 22.n3

(3)

–21 –21


Additional Example 2B: Solving Two-Step Equations

Solve = 9.y – 4 3


– = 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


Additional Example 2B Continued

Solve = 9.y – 4 3

= 9y – 43

y – 4 = 27

+ 4 + 4 Add to undo subtraction.

y = 31



= 9y – 43(3) (3)


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



Check It Out: Example 2A Continued


+ 8 = 18(4)n4

Subtract to undo addition.

n + 32 = 72

n = 40


Solve + 8 = 18.n4

(4)

–32 –32


Check It Out: Example 2B

Solve = 7.y – 7 2


– = 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


Check It Out: Example 2B Continued

Solve = 7.y – 7 2

= 7y – 72

y – 7 = 14

+ 7 + 7 Add to undo subtraction.

y = 21



= 7y – 72(2) (2)


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems


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


1. Solve – 9 = 3.

A. p = –84

B. p = –42

C. p = 42

D. p = 84


p -7


2. Solve 3r + 46 = –29.

A. r = –20

B. r = –25

C. r = –30

D. r = –35



3. Solve –3.3 = –1.2t + 15.3.

A. t = 15.5

B. t = 14

C. t = 12.5

D. t = 10



4. Solve = 5.

A. v = 31

B. v = 32

C. v = 33

D. v = 34


v + 7 8


5. The membership fee of a DVD rental store is $15. The cost of renting a DVD is $2. If John pays $27, how many DVDs did he rent?

A. 6 DVDs

B. 8 DVDs

C. 12 DVDs

D. 24 DVDs


2-8 Solving Two-Step Equations Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of...

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