Below are your two lines.

Question

What is the solution to this system of equations?

Your combined table for both companies is below:

Kites Purchased:0,1
Total Cost for Kites-R-Fun:10,16
Total Cost for Windy Kites:16,20

Answer 1

To solve the system of equations based on the provided data, we first need to interpret the information given about the kites purchased and total costs for each company.

We have the following information:

1. For Kites-R-Fun:

   • 0 kites purchased: Total Cost = $10
   • 1 kite purchased: Total Cost = $16

2. For Windy Kites:

   • 0 kites purchased: Total Cost = $16
   • 1 kite purchased: Total Cost = $20

Step 1: Formulate the Equations

Kites-R-Fun:

Let \( x \) be the number of kites purchased from Kites-R-Fun. The total cost can be represented as:

• If \( x = 0 \), Cost = 10
• If \( x = 1 \), Cost = 16

The cost increase when buying 1 more kite from Kites-R-Fun is: \[ 16 = 10 + (price_per_kite \times 1) \] From this, we can see that the price per kite for Kites-R-Fun is $6.

Thus, the equation for Kites-R-Fun can be expressed as: \[ \text{Total Cost (Kites-R-Fun)} = 10 + 6x \]

Windy Kites:

Let \( y \) be the number of kites purchased from Windy Kites. The total cost can be represented as:

• If \( y = 0 \), Cost = 16
• If \( y = 1 \), Cost = 20

The cost increase when buying 1 more kite from Windy Kites is: \[ 20 = 16 + (price_per_kite \times 1) \] From this, we can see that the price per kite for Windy Kites is $4.

Thus, the equation for Windy Kites can be expressed as: \[ \text{Total Cost (Windy Kites)} = 16 + 4y \]

Step 2: Set Up the System of Equations

Now we have a system of equations for total costs based on the number of kites:

• For Kites-R-Fun: \( C_1 = 10 + 6x \)
• For Windy Kites: \( C_2 = 16 + 4y \)

Step 3: Solve the System

Since the total number of kites purchased (combined from both companies) isn’t provided, we need to form an equation based on combined costs. However, the problem doesn't specify a target cost or context for how or when to combine the total costs.

As an example, if we assume we want to determine a point at which both costs are equal, we could set the equations equal to each other and solve for \( x \) and \( y \):

Setting the equations equal: \[ 10 + 6x = 16 + 4y \]

Simplifying: \[ 6x - 4y = 6 \] or \[ 3x - 2y = 3 \]

Finding Pairs of (x, y):

To find integer solutions, we can express \( y \) in terms of \( x \): \[ 2y = 3x - 3 \implies y = \frac{3x - 3}{2} \]

Since \( y \) must be an integer, \( 3x - 3 \) must be even, which means \( x \) must be odd. Therefore, we can choose odd numbers for \( x \) to find valid pairs \( (x, y) \):

1. If \( x = 1 \):
   \( y = \frac{3(1) - 3}{2} = 0 \)
   Pair \( (1, 0) \) is valid.

That’s one solution to the equations given the context.

Now we can check:

• If we buy 1 kite from Kites-R-Fun and 0 from Windy Kites:
  • Total Cost from Kites-R-Fun: $10 + $6(1) = $16
  • Total Cost from Windy Kites: $16 + $4(0) = $16
  • Both costs equal.

Conclusion

The solution gives us the pairs \( (x, y) = (1, 0) \), meaning you could buy 1 kite from Kites-R-Fun and 0 from Windy Kites, resulting in equal total costs of $16.

Additional possible solutions depend on the maximum number of kites you are allowed to purchase, as per the context not provided in the question.