2. Warm Up Solve. 1. x + 12 = 35 2. 8 x = 120 3. = 7 4. –34 = y + 56 x = 23 x = 15 y = 63 y = –90 y 9
3. AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. California Standards
4. 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.
5. 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
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7. 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. Additional Example 1 Continued 2 Make a Plan
8. 653.05 = 443.75 + 45.50 h – 443.75 –443.75 209.30 = 45.50 h 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. Since 443.75 is added to both sides, subtract 443.75 from both sides. Solve 3 209.30 45.50 h 45.50 45.50 =
9. You can use a table to decide whether your answer is reasonable. Additional Example 1 Continued 4.6 hours is a reasonable answer. Look Back 4 $671.25 $443.75 $227.50 5 $625.75 $443.75 $182.00 4 $580.25 $443.75 $136.50 3 $534.75 $443.75 $91.00 2 $489.25 $443.75 $45.50 1 Total Cost Parts Labor Hours
10. 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
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12. 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. Check It Out! Example 1 Continued 2 Make a Plan
13. 850 = 275 + 35 h – 275 –275 575 = 35 h 16.4 h The mechanic worked for about 16.4 hours on your car. Check It Out! Example 1 Continued Since h is multiplied by 35, divide both sides by 35. Since 275 is added to both sides, subtract 275 from both sides. Solve 3 575 35 h 35 35 =
14. Check It Out! Example 1 Continued You can use a table to decide whether your answer is reasonable. 16.4 hours is a reasonable answer. Look Back 4 $870 $275 $595 17 $835 $275 $560 16 $800 $275 $525 15 $765 $275 $490 14 $730 $275 $455 13 Total Cost Parts Labor Hours
15. Additional Example 2A: Solving Two-Step Equations Solve + 7 = 22. Since n is divided by 3, multiply both sides by 3. n = 45 Method 1: Use fraction operations. n 3 + 7 – 7 = 22 – 7 n 3 3 = 3 15 n 3 + 7 = 22 n 3 Since 7 is added to , subtract 7 from both sides to undo the addition. n3 = 15 n 3
16. Additional Example 2B: Solving Two-Step Equations Solve = 9. 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. Method 2: Multiply both sides of the equation by the denominator. y – 4 3 = 9 y – 4 3 = 9 y – 4 3 (3) (3)
17. Check It Out! Example 2A Solve + 8 = 18. Since n is divided by 4, multiply both sides by 4. n = 40 Method 1: Use fraction operations. n 4 + 8 – 8 = 18 – 8 n 4 4 = 4 10 n 4 + 8 = 18 n 4 Since 8 is added to , subtract 8 from both sides to undo the addition. n4 = 10 n 4
18. Check It Out! Example 2B Solve = 7. 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. Method 2: Multiply both sides of the equation by the denominator. y – 7 2 = 7 y – 7 2 = 7 y – 7 2 (2) (2)
19. Solve. 1. – x – 3 = 10 2. 7 y + 25.68 = –26.12 3. –8.3 = –3.5 x + 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 Quiz y = –7.4 x = –117 x = 6.2 y = 29.1 24 months 1 9 y + 5 11
