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Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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Warm Up Solve. 1. x + 12 = 35 2. 8x = 120 = 7 4. –34 = y + 56 x = 23 x = 15 y 9 y = 63 y = –90

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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

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**Learn to solve two-step equations.**

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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.

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**Additional Example 1: Problem Solving Application**

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?

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**Understand the Problem**

Additional Example 1 Continued 1 Understand the Problem List the important information: The answer is the number of hours the mechanic worked on the car. The parts cost $443. The labor cost $45 per hour. The total bill was $650. Let h represent the hours the mechanic worked. Total bill = Parts + Labor = h

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**Additional Example 1 Continued**

2 Make a Plan 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.

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**Additional Example 1 Continued**

Solve 3 650 = h –443 – Step 1: Subtract to undo the addition. 207 = h h = Step 2: Divide to undo multiplication. 4.6 = h The mechanic worked for 4.6 hours on Mr. Wong’s car.

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**Additional Example 1 Continued**

Look Back 4 You can use a table to decide whether your answer is reasonable. $668 $443 225 5 $623 180 4 $578 135 3 $533 90 2 $488 45 1 Total Cost Parts Labor Hours 4.6 hours is a reasonable answer.

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Check It Out: Example 1 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?

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**Understand the Problem**

Check It Out: Example 1 Continued 1 Understand the Problem List the important information: The answer is the number of hours the mechanic worked on your car. The parts cost $275. The labor cost $35 per hour. The total bill was $850. Let h represent the hours the mechanic worked. Total bill = Parts + Labor 850 = h

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**Check It Out: Example 1 Continued**

2 Make a Plan 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.

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**Check It Out: Example 1 Continued**

Solve 3 850 = h –275 – Step 1: Subtract to undo the addition. 575 = h h = Step 2: Divide to undo multiplication. 16.4 h The mechanic worked for about 16.4 hours on your car.

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**Check It Out: Example 1 Continued**

Look Back 4 You can use a table to decide whether your answer is reasonable. $870 $275 595 17 $835 560 16 $800 525 15 $765 490 14 $730 455 13 Total Cost Parts Labor Hours 16.4 hours is a reasonable answer.

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**Additional Example 2A: Solving Two-Step Equations**

Solve = 22. Method 1: Work backward to isolate the variable. Think: First the variable is divided by 3, and then 7 is added. To isolate the variable, subtract 7, and then multiply by 3. + 7 – 7 = 22 – 7 n 3 Subtract 7 from both sides. 3 = 3 15 n 3 Multiply both sides by 3. n = 45

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**Additional Example 2A Continued**

Solve = 22. Method 2: Multiply both sides of the equation by the denominator. + 7 = 22(3) n 3 (3) Multiply both sides by the denominator. n + 21 = 66 Subtract to undo addition. –21 –21 n = 45

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**Additional Example 2B: Solving Two-Step Equations**

y – 4 3 Solve = 9. Method 1: Work backward to isolate the variable. – = 9 y 3 4 3 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. 4 3 Add to both sides. 4 3 4 3 – = 9 + y 3 (3) = (3) y 3 31 t3 Multiply both sides by 3. y = 31

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**Additional Example 2B Continued**

y – 4 3 Solve = 9. Method 2: Multiply both sides of the equation by the denominator. = 9 y – 4 3 = 9 y – 4 3 (3) (3) Multiply both sides by the denominator. y – 4 = 27 Add to undo subtraction. y = 31

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Check It Out: Example 2A n 4 Solve = 18. Method 1: Work backward to isolate the variable. Think: First the variable is divided by 4, and then 8 is added. To isolate the variable, subtract 8, and then multiply by 4. + 8 – 8 = 18 – 8 n 4 Subtract 8 from both sides. 4 = 4 10 n 4 Multiply both sides by 4. n = 40

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**Check It Out: Example 2A Continued**

Solve = 18. Method 2: Multiply both sides of the equation by the denominator. + 8 = 18(4) n 4 (4) Multiply both sides by the denominator. n + 32 = 72 Subtract to undo addition. –32 –32 n = 40

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Check It Out: Example 2B y – 7 2 Solve = 7. Method 1: Work backward to isolate the variable. – = 7 y 2 7 2 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. 7 2 Add to both sides. 7 2 7 2 – = 7 + y 2 (2) = (2) y 2 21 t2 Multiply both sides by 2. y = 21

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**Check It Out: Example 2B Continued**

y – 7 2 Solve = 7. Method 2: Multiply both sides of the equation by the denominator. = 7 y – 7 2 = 7 y – 7 2 (2) (2) Multiply both sides by the denominator. y – 7 = 14 Add to undo subtraction. y = 21

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Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

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Lesson Quiz Solve. – 3 = 10 2. 7y + 25 = –24 3. –8.3 = –3.5x = 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? x –9 x = –117 y = –7 x = 6.2 y y = 28 24

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**Lesson Quiz for Student Response Systems**

1. Solve – 9 = 3. A. p = –84 B. p = –42 C. p = 42 D. p = 84

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**Lesson Quiz for Student Response Systems**

2. Solve 3r + 46 = –29. A. r = –20 B. r = –25 C. r = –30 D. r = –35

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**Lesson Quiz for Student Response Systems**

3. Solve –3.3 = –1.2t A. t = 15.5 B. t = 14 C. t = 12.5 D. t = 10

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**Lesson Quiz for Student Response Systems**

v 4. Solve = 5. A. v = 31 B. v = 32 C. v = 33 D. v = 34

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**Lesson Quiz for Student Response Systems**

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

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11-1 Solving Two-Step Equations When you solve equations that have one operation, you use an inverse operation to isolate the variable. n + 7 = 15 – 7.

11-1 Solving Two-Step Equations When you solve equations that have one operation, you use an inverse operation to isolate the variable. n + 7 = 15 – 7.

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