California Standards 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.

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.

Additional Example 1: Problem Solving Application 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?

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 $443.75. The labor cost $45.50 per hour. The total bill was $653.05. Let h represent the hours the mechanic worked. Total bill = Parts + Labor 653.05 = 443.75 + 45.50h

Additional Example 1 Continued 2 Make a Plan 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 Solve 3 Since 443.75 is added to both sides, subtract 443.75 from both sides. 653.05 = 443.75 + 45.50h –443.75 –443.75 209.30 = 45.50h Since h is multiplied by 45.50, divide both sides by 45.50. 209.30 45.50h 45.50 45.50 = 4.6 = h The mechanic worked for 4.6 hours on Mr. Wong’s car.

Additional Example 1 Continued Look Back 4 You can use a table to decide whether your answer is reasonable. Hours Labor Parts Total Cost 1 $45.50 $443.75 $489.25 2 $91.00 $534.75 3 $136.50 $580.25 4 $182.00 $625.75 5 $227.50 $671.25 4.6 hours is a reasonable answer.

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?

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 = 275 + 35h

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.

Check It Out! Example 1 Continued Solve 3 Since 275 is added to both sides, subtract 275 from both sides. 850 = 275 + 35h –275 –275 575 = 35h Since h is multiplied by 35, divide both sides by 35. 575 35h 35 35 = 16.4  h The mechanic worked for about 16.4 hours on your car.

Check It Out! Example 1 Continued Look Back 4 You can use a table to decide whether your answer is reasonable. Hours Labor Parts Total Cost 13 $455 $275 $730 14 $490 $765 15 $525 $800 16 $560 $835 17 $595 $870 16.4 hours is a reasonable answer.

Additional Example 2A: Solving Two-Step Equations Solve + 7 = 22. Method 1: Use fraction operations. + 7 = 22 n3 Since 7 is added to , subtract 7 from both sides to undo the addition. n3 + 7 – 7 = 22 – 7 n 3 = 15 n3 3  = 3  15 n3 Since n is divided by 3, multiply both sides by 3. n = 45

Additional Example 2B: Solving Two-Step Equations 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 + 4 + 4 Since 4 is subtracted from y, add 4 to both sides to undo the subtraction. y = 31

Check It Out! Example 2A n4 Solve + 8 = 18. Method 1: Use fraction operations. + 8 = 18 n4 Since 8 is added to , subtract 8 from both sides to undo the addition. n4 + 8 – 8 = 18 – 8 n 4 = 10 n4 4  = 4  10 n4 Since n is divided by 4, multiply both sides by 4. n = 40

Check It Out! Example 2B 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 + 7 + 7 Since 7 is subtracted from y, add 7 to both sides to undo the subtraction. y = 21