Question

Algebra 2 Word Problem: Set up using 3 variables & 3 equations:
"A friend e-mails you the results of a recent high school swim meet. The e-mail states that 24 individuals placed, earning a combined total of 53 points. First place earned 3 points, second place earned 2 points, and third place earned 1 point. There were as many first-place finishers as second and third-place finishers combined."
A: Write a system of THREE equations that represents how many people finished in each place.
B: How many swimmers finished in first place, second and third?

Answer 1

A) The answer is
x + y + z = 24
3x + 2y + z = 53
x = y + z


x - the number of swimmers in the first place
y - the number of swimmers in the second place
z - the number of swimmers in the third place

1. The e-mail states that 24 individuals placed: x + y + z = 24
2. First place earned 3 points, second place earned 2 points, and third place earned 1 point, earning a combined total of 53 points: 3x + 2y + z = 53
3.  There were as many first-place finishers as second and third-place finishers combined: x = y + z

The system of three equations is:
x + y + z = 24
3x + 2y + z = 53
x = y + z



B) The answer is
12 swimmers in the first place
5 swimmers in the second place
7 swimmers in the third place


(i) x + y + z = 24
(ii) 3x + 2y + z = 53
(iii) x = y + z
______
Substitute y + z from the third equation into the first one:
(i) x + y + z = 24
(iii) x = y + z
______
x + x = 24
2x = 24
x = 24 / 2
x = 12
_____

x = y + z
x = 12
y + z = 12
z = 12 - y

3x + 2y + z = 53
3 * 12 + 2y + (12 - y) = 53
36 + 2y + 12 - y = 53
2y - y + 36 + 12 = 53
y + 48 = 53
y = 53 - 48
y = 5

z = 12 - y
z = 12 - 5
z = 7