Download - Solving Systems by Graphing
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Solving Systems by Graphing
Please view this tutorial and answer the follow-up questions on loose leaf to
turn in to your teacher.
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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.
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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.
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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
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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
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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
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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?
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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.
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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
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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.
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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.
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ExamplePlanet Fitness: C = 15 + 10M
Bally’s: C = 35 + 8M
Next, set your window so that you can see the intersection point.
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ExamplePlanet Fitness: C = 15 + 10M
Bally’s: C = 35 + 8M
Hit GRAPH. The intersection point will be
your solution.
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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.
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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.)
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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.
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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.
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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?
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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.
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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.
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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.
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ExamplePlanet Fitness: C = 15 + 10M
Bally’s: C = 35 + 8M
X = 10 and Y = 115
You could also answer these questions using a
graph.
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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.
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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
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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.
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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.
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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?