Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Mr. Boyer is the volunteer coordinator for a company that puts on running races. Last year, the company organized 12 short races and 10 long races, which required a total of 1,168 volunteers. This year, the company organized 12 short races and 9 long races, which required 1,086 volunteers in total. How many volunteers does each type of race require?

It takes
volunteers for a short race, and
volunteers for a long race.

Answer 1

Let \( x \) be the number of volunteers needed for a short race, and \( y \) be the number of volunteers needed for a long race.

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

1. For last year with 12 short races and 10 long races:
\[
12x + 10y = 1168
\]

2. For this year with 12 short races and 9 long races:
\[
12x + 9y = 1086
\]

Now we have the system of equations:
\[
\begin{cases}
12x + 10y = 1168 \quad \text{(1)} \\
12x + 9y = 1086 \quad \text{(2)}
\end{cases}
\]

To solve this system, we can use the elimination method. First, we'll subtract equation (2) from equation (1).

\[
(12x + 10y) - (12x + 9y) = 1168 - 1086
\]

This simplifies to:
\[
10y - 9y = 1168 - 1086
\]
\[
y = 1168 - 1086
\]
\[
y = 82
\]

Now that we have \( y = 82 \), we can substitute this value back into one of the original equations to find \( x \). We'll use equation (2):

\[
12x + 9(82) = 1086
\]

Now calculate \( 9(82) \):
\[
12x + 738 = 1086
\]

Subtract 738 from both sides:
\[
12x = 1086 - 738
\]
\[
12x = 348
\]

Now, divide by 12:
\[
x = \frac{348}{12} = 29
\]

Thus, the solution to the system is:
- \( x = 29 \) (volunteers needed for a short race)
- \( y = 82 \) (volunteers needed for a long race)

Finally, we fill in the blanks:

It takes **29** volunteers for a short race, and **82** volunteers for a long race.