This document summarizes steps for solving equations with variables on both sides, including using addition or subtraction to isolate the variable on one side and then simplifying. It provides an example of solving the equation 9a + 2 = 4a - 18 for a = -4. The document also applies these steps to a real-world problem about two bicyclists traveling at different rates and calculating the time for one to catch up to the other. It presents the equation to model the problem and solves for the time.

1. Chapter 7 Section 5: Solving Equations with Variables on Both Sides. January 8 th , 2009

2. Warm Up: 10(5 + m) = 63 -x/6 – 7 = 0 0.6x + 1.9x = 5 1.3 = m -42 = x 2 = x

3. Variables On Both Sides? Use addition or subtraction to get the variable on only one side of the equation. 9a + 2 = 4a – 18 - 4a -4a Sub. Prop. of Equality 5a + 2 = -18 Simplify - 2 - 2 Sub. Prop. of Equality 5a = - 20 Simplify 5 5 Div. Prop. of Equality a = -4 Solved NOW CHECK!

4. Check Your Answer! 9 a + 2 = 4 a – 18 for a = -4 9( -4 ) + 2 = 4( -4 ) – 18 -36 + 2 = -16 – 18 -34 = -34 Correct!

5. Try These 4x + 4 = 2x + 36 -15 + 6b = -8b + 13 16 = x 2 = b

6. Using Equations with Variables on BOTH Sides. Use the Distributive Property to simplify one or both sides of an equation before the variable is alone on one side.

7. Real World Experiences Beth leaves home on her bicycle, riding at a steady rate of 8 miles/hour. Her brother Ted leaves home on his bicycle half an hour later, following Beth’s route. He rides at a steady rate of 12 miles/hour. How long after Beth leaves home will Ted catch up to her ? Beth’s Distance Traveled = Ted’s Distance Traveled 8 mi/h • Beth’s Time = 12 mi/h • Ted’s Time If X = Beth’s Time, Then (X – ½) = Ted’s Time. Because Ted left ½ hour after Beth.

8. Put It All Together Beth’s Distance Traveled = Ted’s Distance Traveled 8 mi/h • Beth’s Time = 12 mi/h • Ted’s Time If X = Beth’s Time, Then (X – ½) = Ted’s Time. Because Ted left ½ hour after Beth. 8 mi/h (X) = 12 mi/h (X – ½) 8 X = 12 (X – ½) 8 X = 12X – 6 -4 X = -6 X = 6/4 = 1 ½ Hours 1 ½ hours after Beth leaves is when her brother, Ted, catches up with her.

9. Try This One Yourselves CAR A leaves Eastown traveling at a steady rate of 50 miles/hour. CAR B leaves Eastown 1 hour later following CAR A. It travels at a steady rate of 60 miles/hour. How long after CAR A leaves Eastown will CAR B catch up? 6 Hours after CAR A leaves is when CAR B catches up.

10. Assignment #55 Pages 357-358: 1-28 all. Begin In Class