Question

The Cherry Creek Middle School band held a car wash to raise money for new uniforms. The band charged $10 for cars and $15 for larger vehicles, like vans or SUVs. In the first hour of the car wash, band members washed a total of 10 vehicles and raised $120 toward their uniforms.
Graph a system of equations that represents this story.
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Cars washed
Larger vehicles washed
Car wash fundraiser
vehiclesdollars

Answer 1

To represent the scenario described in the Cherry Creek Middle School band car wash fundraiser, we can define two variables:

- Let \( x \) be the number of cars washed.
- Let \( y \) be the number of larger vehicles (vans or SUVs) washed.

From the information provided, we can set up the following system of equations:

1. The first equation represents the total number of vehicles washed:
\[
x + y = 10
\]

2. The second equation represents the total amount of money raised:
\[
10x + 15y = 120
\]

Now, we can graph these equations.

**To graph the equations:**

1. **Graph the first equation \( x + y = 10 \):**
- Rewrite it in slope-intercept form (if you want to plot easily):
\[
y = -x + 10
\]
- This line intersects the y-axis at (0, 10) and the x-axis at (10, 0).

2. **Graph the second equation \( 10x + 15y = 120 \):**
- Rewrite it in slope-intercept form:
\[
15y = -10x + 120
\]
\[
y = -\frac{2}{3}x + 8
\]
- This line intersects the y-axis at (0, 8) and the x-axis at (12, 0) (as it can be found by setting \( y = 0 \)).

**Graphing Steps:**

- Plot the y-intercept and x-intercept of both lines on the grid.
- For \( x + y = 10 \), plot (0, 10) and (10, 0).
- For \( 10x + 15y = 120 \), plot (0, 8) and (12, 0).
- Draw the lines through these points.

Where these two lines intersect will give you the solution to the system of equations, which represents the values of \( x \) (cars washed) and \( y \) (larger vehicles washed).

To find the intersection algebraically:

1. From \( x + y = 10 \), we can express \( y \) as:
\[
y = 10 - x
\]

2. Substitute \( y \) into the second equation:
\[
10x + 15(10 - x) = 120
\]
\[
10x + 150 - 15x = 120
\]
\[
-5x + 150 = 120
\]
\[
-5x = -30
\]
\[
x = 6
\]

3. Substitute \( x = 6 \) back into \( y = 10 - x \):
\[
y = 10 - 6 = 4
\]

Thus, the solution is \( x = 6 \) (cars washed) and \( y = 4 \) (larger vehicles washed).

You can plot these two lines on a graph, and where they intersect at the point (6, 4) illustrates how many of each type of vehicle was washed.