To solve this system of equations, we will analyze the data provided for both companies, K-R-F and Windy Kites, by determining the price per kite for each company and plotting them.
We have the following data:
For K-R-F:
- Kites Purchased (x): 0, 1, 4, 7, 10
- Total Cost (y): $10, $16, $34, $52, $70
For Windy Kites:
- Kites Purchased (x): 0, 1, 4, 7, 10
- Total Cost (y): $16, $20, $32, $44, $56
Step 1: Determine the Price per Kite for Each Company
For K-R-F:
- From 0 to 1 kite: \( \frac{16 - 10}{1 - 0} = 6 \) (Price per kite is $6 after the first purchase)
- From 1 to 4 kites: \( \frac{34 - 16}{4 - 1} = \frac{18}{3} = 6 \)
- From 4 to 7 kites: \( \frac{52 - 34}{7 - 4} = \frac{18}{3} = 6 \)
- From 7 to 10 kites: \( \frac{70 - 52}{10 - 7} = \frac{18}{3} = 6 \)
So, K-R-F charges a constant price of $6 per kite after the initial cost.
For Windy Kites:
- From 0 to 1 kite: \( \frac{20 - 16}{1 - 0} = 4 \) (Price per kite is $4 after the first purchase)
- From 1 to 4 kites: \( \frac{32 - 20}{4 - 1} = \frac{12}{3} = 4 \)
- From 4 to 7 kites: \( \frac{44 - 32}{7 - 4} = \frac{12}{3} = 4 \)
- From 7 to 10 kites: \( \frac{56 - 44}{10 - 7} = \frac{12}{3} = 4 \)
Windy Kites charges a constant price of $4 per kite after the initial cost.
Step 2: Create the Equations
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K-R-F's total cost equation can be expressed as: \[ y = 6x + 10 \]
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Windy Kites' total cost equation can be expressed as: \[ y = 4x + 16 \]
Step 3: Solve the System of Equations
To find the intersection where both total costs are the same, set the two equations equal to each other:
\[ 6x + 10 = 4x + 16 \]
Now, solve for \(x\): \[ 6x - 4x = 16 - 10 \] \[ 2x = 6 \] \[ x = 3 \]
Now substitute \(x = 3\) back into either equation to find \(y\): Using K-R-F: \[ y = 6(3) + 10 = 18 + 10 = 28 \]
Therefore, the solution to this system of equations is the point (3, 28).
Final Answer: (3, 28)