SOLUTION: I am trying to help my daughter with her Algebra homework and it has been years since I've done a problem of this type. I would appreciate it if you could give me some direction o

Question 108713: I am trying to help my daughter with her Algebra homework and it has been years since I've done a problem of this type. I would appreciate it if you could give me some direction on how to solve this word problem:
"Aaron, Beth, Candace, Darvel and Emil ran around a track as part of a fundraiser for their school. Aaron ran first, and then when he was tired, Beth ran. When Beth got tired, Candace ran, and so on. By the end of the fundraiser, they had run a total of 30 miles. Aaron and Beth combined ran 40 percent of the total distance. Beth and Candace combined ran 34 percent of the total distance. Candace and Darvel combined ran 40 percent of the total distance. And Darvel and Emil combined ran 41 percent of the total distance. How far did each person run?"
This problem was on a worksheet from the teacher that said "Solve It Problem!" at the top of the page. My daughter thinks her teacher got it from a math magazine but we're not sure which one. Please help me help her get this solved. Thank you so much!
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Aaron, Beth Candance, John and Pat ran around a track as part of a fundraiser for their school Aaron ran first and then when he was tired, Beth ran. When Beth got tired, Candace ran, and so on. By the end of the fundraiser,
they had run a total of 30 miles.
Aaron and Beth combined ran 40% of the total distance.
A + B = 12 miles
Beth and Candace combined ran 34% of the total distance.
B + C = 10.2 miles
Candace and John combined ran 40% of the total distance.
C + J = 12 miles
John and Pat combined ran 41% of the total distance.
J + P = 12.3 miles
How far did each person run?
--------------------
A + B + C + J + P = 30
A + (10.2-C) + C + J + (12.3-J) = 30
A + 10.2 + 12.3 = 30
A = 7.5 miles
B = 12 - 7.5 = 4.5 miles
C = 10.2 - 4.5 = 5.7 miles
J = 12 - 5.7 ] = 6.3 miles
P = 12.3 - 6.3 = 6 miles
=================
Cheers,
Stan H.

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Let's use a, b, c, d, and e to represent the distances that each of the students ran.
:
We are given several relationships, so let's express them symbolically:
:
:Recall that the total overall distance was 30 miles.
:
:
: and finally,
:
:
We now have 5 linear equations in 5 variables. So let's start with the augmented matrix and perform Gaussian Elimination.
:

:
Subtract row 2 from row 1 and place the result in row 2:
:

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