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.

Question

Graph a system of equations that represents this story.
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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.