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What is the solution to this system of equations?

(,)
Graph my answer
Your combined table for both companies is below:

Kites Purchased
Total Cost for Kites-R-Fun
Total Cost for Windy Kites
0
$10
$16
1
$16
$20
4
$34
$32
7
$52
$44
10
70
$​56

1 answer

To solve the system of equations represented by the data in the table, we need to find the equations that describe the total cost for each company based on the number of kites purchased. Let's denote the number of kites purchased by \( x \), the total cost for Kites-R-Fun by \( y_1 \), and the total cost for Windy Kites by \( y_2 \).

Based on the table, we can analyze the cost data:

Kites-R-Fun:

From the table:

  • At 0 kites: $10
  • At 1 kite: $16
  • At 4 kites: $34
  • At 7 kites: $52
  • At 10 kites: $70

Let's determine a linear equation for Kites-R-Fun. The total cost appears to increase linearly. We can choose two points from the data to derive the slope:

Using the points (0, 10) and (10, 70):

  1. The slope \( m \) can be calculated as: \[ m = \frac{70 - 10}{10 - 0} = \frac{60}{10} = 6 \]
  2. The equation in slope-intercept form \( y = mx + b \) becomes: \[ y_1 = 6x + 10 \]

Windy Kites:

From the table:

  • At 0 kites: $16
  • At 1 kite: $20
  • At 4 kites: $32
  • At 7 kites: $44
  • At 10 kites: $56

For Windy Kites, we can also choose two points to derive the slope: Using the points (0, 16) and (10, 56):

  1. The slope \( m \) can be calculated as: \[ m = \frac{56 - 16}{10 - 0} = \frac{40}{10} = 4 \]
  2. The equation in slope-intercept form becomes: \[ y_2 = 4x + 16 \]

Setting the equations equal to find the intersection:

We now have our system of equations:

  1. \( y_1 = 6x + 10 \) (for Kites-R-Fun)
  2. \( y_2 = 4x + 16 \) (for Windy Kites)

Setting them equal to find where the costs are the same: \[ 6x + 10 = 4x + 16 \] Subtract \( 4x \) from both sides: \[ 2x + 10 = 16 \] Subtract 10 from both sides: \[ 2x = 6 \] Divide by 2: \[ x = 3 \]

Finding the total cost at \( x = 3 \):

Using either equation to find the total cost: Using Kites-R-Fun: \[ y_1 = 6(3) + 10 = 18 + 10 = 28 \]

Solution:

The solution to the system is \( (3, 28) \), meaning that both companies will charge the same amount of $28 when 3 kites are purchased.

Graphing the equations:

To graph, plot:

  • The line for \( y_1 = 6x + 10 \) for Kites-R-Fun.
  • The line for \( y_2 = 4x + 16 \) for Windy Kites.
  • Mark the intersection point \( (3, 28) \).

If you need assistance creating the graph, let me know and I can provide further guidance!