1. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers AF4.1 Solve two-step linear equations in one variable over the rational numbers and verify the reasonableness of the results. California Standards

2. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers Two-step equations contain two operations, and therefore, require two inverse operations to solve. One method to solve the equation is to work backward to undo the operations. Notes

3. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 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 1.Numbers 2.Understand 3.Plan 4.Solve

4. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 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 $443.75. The labor cost $45.50 per hour. The total bill was $653.05. Total bill=Parts +Labor 653.05 = 443.75 + 45.50h

5. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 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. 2 Make a Plan Additional Example 1 Continued

6. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 653.05 = 443.75 + 45.50h Solve 3 –443.75 209.30 = 45.50h 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. 209.30 45.50h 45.50 = Since 443.75 is added to both sides, subtract 443.75 from both sides.

7. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 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 Elbow Partner 1.Numbers 2.Understand 3.Plan 4.Solve

8. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 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

9. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 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 Check It Out! Example 1 Continued

10. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers 850 = 275 + 35h Solve 3 –275 575 = 35h 16.4  h The mechanic worked for about 16.4 hours on your car. Check It Out! Example 1 Continued 575 35h 35 = Since h is multiplied by 35, divide both sides by 35. Since 275 is added to both sides, subtract 275 from both sides.

11. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers Additional Example 2A: Solving Two-Step Equations Solve + 7 = 22. n3n3 + 7 – 7 = 22 – 7 n3n3 Since n is divided by 3, multiply both sides by 3. 3  = 3  15 n3n3 n = 45 Method 1: Use fraction operations. + 7 = 22 n3n3 Since 7 is added to, subtract 7 from both sides to undo the addition. n3n3 = 15 n3n3 Notes

12. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers Additional Example 2B: Solving Two-Step Equations Solve = 9. y – 4 3 = 9 y – 4 3 y – 4 = 27 + 4 + 4Since 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. = 9 y – 4 3 (3) Notes

13. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers Check It Out! Example 2A Solve + 8 = 18. n4n4 + 8 – 8 = 18 – 8 n4n4 Since n is divided by 4, multiply both sides by 4. 4  = 4  10 n4n4 n = 40 Method 1: Use fraction operations. + 8 = 18 n4n4 Since 8 is added to, subtract 8 from both sides to undo the addition. n4n4 = 10 n4n4 Rally Coach A

14. Evaluating Algebraic Expressions 2-8 Two-Step Equations with Rational Numbers Check It Out! Example 2B Solve = 7. y – 7 2 = 7 y – 7 2 y – 7 = 14 + 7 + 7Since 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. = 7 y – 7 2 (2) Rally Coach B