Question

Jimmy's Sports Apparel sells all of their hats for $15.95 each. The cost to produce the hats includes a one-time fixed cost of $75 and $10.95 per each hat. The revenue, or income received by selling hats, versus the cost of producing hats, is represented on the graph below.

A line graph of cost and revenue of quantity of hats verses price in dollars whose intersection is marked with dotted lines.

The solution to the system of equations graphed above occurs when the revenue
.

The solution has an x-value of
and a y-value of
.

Answer 1

To find the point of intersection (the solution to the system of equations) between the revenue and cost lines, we can set up the equations for revenue and cost based on the information provided.

**Revenue Equation:**
The revenue (R) from selling hats is given by:
\[ R = 15.95x \]
where \( x \) is the number of hats sold.

**Cost Equation:**
The cost (C) of producing hats is represented by:
\[ C = 75 + 10.95x \]

**Finding the Intersection:**
To find the point where revenue equals cost (i.e., \( R = C \)), we set the equations equal to each other:
\[ 15.95x = 75 + 10.95x \]

Now, we can solve for \( x \):
1. Subtract \( 10.95x \) from both sides:
\[
15.95x - 10.95x = 75
\]
\[
5x = 75
\]
2. Divide both sides by 5:
\[
x = 15
\]

Now that we have the x-value, we can find the corresponding y-value (either cost or revenue works, as they should be equal at the intersection):
Using the revenue equation:
\[
R = 15.95 \times 15 = 239.25
\]
or using the cost equation:
\[
C = 75 + 10.95 \times 15 = 75 + 164.25 = 239.25
\]

Thus, the solution to the system of equations occurs when the revenue equals the cost:

- The x-value (quantity of hats) is **15**
- The y-value (revenue or cost) is **239.25**

Therefore:

The solution to the system of equations occurs when the revenue equals the cost, with an x-value of **15** and a y-value of **239.25**.