Solving Systems by Graphing

Please view this tutorial and answer the follow-up questions on loose leaf to

turn in to your teacher.

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.

ExampleChris 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.

ExampleThe 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: $15Change: $10

ExampleThe 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: $15Change: $10

ExampleThe 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: $15Change: $10

ExampleLet’s look at Bally’s next.

Bally’s costs $35 to join plus $8 per month.

Starting Point: $35Change: $8

What is your starting point and how does the cost change per month?

ExampleLet’s look at Bally’s next.

Bally’s costs $35 to join plus $8 per month.

Starting Point: $35Change: $8

Now that you have these values, write an equation to model the

cost for Bally’s.

ExampleLet’s look at Bally’s next.

Bally’s costs $35 to join plus $8 per month.

Starting Point: $35Change: $8

C = Cost and M = # of months

C = 35 + 8M

ExampleSince 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.

ExamplePlanet 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.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

Next, set your window so that you can see the intersection point.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

Hit GRAPH. The intersection point will be

your solution.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

To find this point, hit 2nd TRACE then go down to 5:Intersect. Hit Enter.

ExamplePlanet 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.)

ExamplePlanet 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.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

What do these numbers mean?

At 10 months, the cost for both gyms is $115.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

When will it be cheaper to join Planet Fitness? When will it be cheaper to join Bally’s?

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

Use a table to determine when each gym will be cheaper.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

For less than 10 months, you should join Planet Fitness because the cost

would be cheaper than Bally’s.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

For more than 10 months, you should

join Bally’s because the cost would be cheaper

than Planet Fitness.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

You could also answer these questions using a

graph.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

For less than 10 months, look to see

which line is below the other.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

Here, the line for Planet Fitness is below

the line for Bally’s. Planet Fitness

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

For more than 10 months, look to see

which line is below the other.

ExamplePlanet Fitness: C = 15 + 10M

Bally’s: C = 35 + 8M

X = 10 and Y = 115

Bally’s

Here, the line for Bally’s is below the line

for Planet Fitness.

Follow-Up QuestionsJames 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.

1. Make a graph for each equation and find the intersection point.

2. When will both companies charge James the same amount? How much would they charge?

3. When will T-Shirt Time be more economical?4. When will Funny T’s be more economical?