10-1

1. Solving Two-Step Equations 10-1 Warm Up Problem of the Day Lesson Presentation Pre-Algebra

2. Solving Two-Step Equations 10-1 y9 Pre-Algebra Warm Up Solve. 1.x + 12 = 35 2. 8x = 120 3.= 7 4. –34 = y + 56 x = 23 x = 15 y = 63 y = –90

3. Problem of the Day x is an odd integer. If you triple x and then subtract 7, you get a prime number. What is x? (Hint: Think about what the prime number must be in order for x to be an odd.) x = 3

4. OBJECTIVE FOR TODAY Learn to solve two-step equations.

5. 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.

6. 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?

7. 1 Understand the Problem Additional Example 1 Continued 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 650 = 443 + 45h

8. Make a Plan 2 Additional Example 1 Continued 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.

9. 3 Solve 207 45h = 4545 Additional Example 1 Continued 650 = 443 + 45h –443–443Subtract to undo the addition. 207 = 45h Divide to undo multiplication. 4.6 = h The mechanic worked for 4.6 hours on Mr. Wong’s car.

10. 4 Look Back Additional Example 1 Continued If the mechanic worked 4.6 hours, the labor would be $45(4.6) = $207. The sum of the parts and the labor would be $443 + $207 = $650.

11. Try This: 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?

12. 1 Understand the Problem Try This: Example 1 Continued 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 = 275 + 35h

13. Make a Plan 2 Try This: Example 1 Continued 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.

14. 3 Solve 575 35h = 3535 Try This: Example 1 Continued 850 = 275 + 35h –275–275Subtract to undo the addition. 575 = 35h Divide to undo multiplication. 16.4 h The mechanic worked for about 16.4 hours on your car.

15. 4 Look Back Try This: Example 1 Continued If the mechanic worked 16.4 hours, the labor would be $35(16.4) = $574. The sum of the parts and the labor would be $275 + $574 = $849.

16. n3 n3 n3 n3 + 7 = 22 = 15 Multiply to undo division. 3 = 3  15 Additional Example 2A: Solving Two-Step Equations Solve. A. + 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. – 7– 7Subtract to undo addition. n = 45

17. n3 + 7 = 22 453 ? + 7 = 22 ? 15 + 7 = 22 Additional Example 2A Continued Check Substitute 45 into the original equation. 

18. –3.9 = –1.3m –1.3 –1.3 Additional Example 2B: Solving Two-Step Equations B. 2.7 = –1.3m + 6.6 Think: First the variable is multiplied by –1.3, and then 6.6 is added. To isolate the variable, subtract 6.6, and then divide by –1.3. 2.7 = –1.3m + 6.6 –6.6–6.6 Subtract to undo addition. –3.9 = –1.3m Divide to undo multiplication. 3 = m

19. C. = 9 y – 4 y – 4 y – 4 3 3 3 = 9 = 9 3 ·3 ·Multiply to undo division. Additional Example 2C: Solving Two-Step Equations Think: First 4 is subtracted from the variable, and then the result is divided by 3. To isolate the variable, multiply by 3, and then add 4. y – 4 = 27 + 4+ 4Add to undo subtraction. y = 31

20. n4 n4 n4 + 5 = 29 Multiply to undo division. 4  = 4  24 Try This: Example 2A Solve. A. + 5 = 29 Think: First the variable is divided by 4, and then 5 is added. To isolate the variable, subtract 5, and then multiply by 4. – 5– 5Subtract to undo addition. n = 96

21. n4 + 5 = 29 964 ? + 5 = 29 ? 24 + 5 = 29 Try This: Example 2A Continued Check Substitute 96 into the original equation. 

22. 4.6 = –2.3m –2.3 –2.3 Try This: Example 2B B. 4.8 = –2.3m + 0.2 Think: First the variable is multiplied by –2.3, and then 0.2 is added. To isolate the variable, subtract 0.2, and then divide by –2.3. 4.8 = –2.3m + 0.2 –0.2–0.2 Subtract to undo addition. 4.6 = –2.3m Divide to undo multiplication. –2 = m

23. C. = 8 y – 2 y – 2 y – 2 4 4 4 = 8 = 8 4 ·4 ·Multiply to undo division. Try This: Example 2C Think: First 2 is subtracted from the variable, and then the result is divided by 4. To isolate the variable, multiply by 4, and then add 2. y – 2 = 32 + 2+ 2Add to undo subtraction. y = 34

24. x –9 y + 5 11 Lesson Quiz 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? x = –117 y = –7 x = 6.2 y = 28 24 months