Solving Systems by Graphing

1. Solving Systems by Graphing Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.

2. Basics • The solution to a system of equations is a pair of (x,y) values that when substituted into the original equations will make both equations true. • When looking at the graph of the system, this pair of (x, y) values will occur at the intersection point.

3. Example Chris decided to join a gym and needs to decide between Planet Fitness and Bally’s. Planet Fitness costs $15 to join plus $10 per month. Bally’s costs $35 to join plus $8 per month. There is no yearly contract for either gym. Write equations for each gym and decide when each would be more economical.

4. Example The first step is to write equations for each gym. Let’s look at Planet Fitness first. Planet Fitness costs $15 to join plus $10 per month. What is your starting point and how does the cost change per month? Starting Point: $15 Change: $10

5. Example The first step is to write equations for each gym. Let’s look at Planet Fitness first. Planet Fitness costs $15 to join plus $10 per month. Now that you have these values, write an equation to model the cost for Planet Fitness. Starting Point: $15 Change: $10

6. Example The first step is to write equations for each gym. Let’s look at Planet Fitness first. Planet Fitness costs $15 to join plus $10 per month. C = Cost and M = # of months C = 15 + 10M Starting Point: $15 Change: $10

7. Example Let’s look at Bally’s next. Bally’s costs $35 to join plus $8 per month. What is your starting point and how does the cost change per month? Starting Point: $35 Change: $8

8. Example Let’s look at Bally’s next. Bally’s costs $35 to join plus $8 per month. Now that you have these values, write an equation to model the cost for Bally’s. Starting Point: $35 Change: $8

9. Example Let’s look at Bally’s next. Bally’s costs $35 to join plus $8 per month. C = Cost and M = # of months C = 35 + 8M Starting Point: $35 Change: $8

10. Example Since we have an equation to model the cost for each gym, we can now graph those equations to find the solution to the system. Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M You will see the solution as the intersection point of the two equations.

11. Example Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M First, enter your equation into the calculator by hitting the Y= button then typing the Planet Fitness equation into Y1 and Bally’s into Y2.

12. Example Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M Next, set your window so that you can see the intersection point.

13. Example Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M Hit GRAPH. The intersection point will be your solution.

14. Example Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M To find this point, hit 2nd TRACE then go down to 5:Intersect. Hit Enter.

15. Example Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M Hit ENTER three times. (The first time tells the calculator the first line, the second tells the second line, the third tells your guess.)

16. Example Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M You’ll see the coordinates of the point of intersection at the bottom of the screen.

17. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M What do these numbers mean? At 10 months, the cost for both gyms is $115.

18. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M When will it be cheaper to join Planet Fitness? When will it be cheaper to join Bally’s?

19. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M Use a table to determine when each gym will be cheaper.

20. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M For less than 10 months, you should join Planet Fitness because the cost would be cheaper than Bally’s.

21. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M For more than 10 months, you should join Bally’s because the cost would be cheaper than Planet Fitness.

22. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M You could also answer these questions using a graph.

23. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M For less than 10 months, look to see which line is below the other.

24. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M Here, the line for Planet Fitness is below the line for Bally’s. Planet Fitness

25. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M For more than 10 months, look to see which line is below the other.

26. Example X = 10 and Y = 115 Planet Fitness: C = 15 + 10M Bally’s: C = 35 + 8M Here, the line for Bally’s is below the line for Planet Fitness. Bally’s

27. Follow-Up Questions James wants to buy a customized t-shirt for his girlfriend for Valentine’s day. T-Shirt Time gives him a quote of $20 for the t-shirt and 0.25 for each letter. Funny T’s has a price of $15 for the shirt plus an additional 0.75 per letter. Make a graph for each equation and find the intersection point. When will both companies charge James the same amount? How much would they charge? When will T-Shirt Time be more economical? When will Funny T’s be more economical?