Question

write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.
mr. mccarthy is the volunteer coordinator for a company that puts on running races. last year, the company organized 6 short races and 6 long races, which required a total of 642 volunteers. this year, the company organized 8 short races and 7 long races, which required 780 volunteers in total. how many volunteers does each type of race require?
it takes (\boxed{quad}) volunteers for a short race, and (\boxed{quad}) volunteers for a long race.

Explanation:

Step1: Define Variables

Let \( x \) be volunteers for short race, \( y \) for long race.

Step2: Form Equations

Last year: \( 6x + 6y = 642 \) (simplify: \( x + y = 107 \))
This year: \( 8x + 7y = 780 \)

Step3: Elimination Method

From \( x + y = 107 \), get \( y = 107 - x \).
Substitute into \( 8x + 7y = 780 \):
\( 8x + 7(107 - x) = 780 \)
\( 8x + 749 - 7x = 780 \)
\( x + 749 = 780 \)
\( x = 780 - 749 = 31 \)

Step4: Find \( y \)

Substitute \( x = 31 \) into \( y = 107 - x \):
\( y = 107 - 31 = 76 \)

Answer:

31 volunteers for a short race, and 76 volunteers for a long race.