matrix word problem

Question

At Edmonds Community College freshman need to take a 100 level Math and English plus optional Philosophy course.

In any quarter the college needs to make available 8 less English sections than Math sections.
In any quarter student demand for the optional Philosophy course is half as many sections as English sections.
Available classrooms limit the total sections of all three courses to 53

Given these constraints how many sections of each course should the college make available each quarter to meet demand?

Answer 1

This problem can be solved with a simple system of equations.

First, we assign each of the classes a variable: English = E, Math = M, and Philosophy = P. Then we translate each of the sentences in the second paragraph into mathematical equations using our variables.

"In any quarter the college needs to make available 8 less English sections than Math sections" becomes

M - 8 = E (or E + 8 = M).

"In any quarter student demand for the optional Philosophy course is half as many sections as English sections" could be represented as E/2 = P (or 2P = E)

And finally "Available classrooms limit the total sections of all three courses to 53" just means

E + M + P = 53

So now we choose two different variables from the first two equations we've created to substitute into the third equation, so we can solve for the single remaining variable. The only variable that appears in all three equations is E, so we'll substitute everything else for E, like so:

E + (E + 8) + (E/2) = 53 --> the original E stays, M from the first equation, P from the second.

Solve (preserving the order of operations), and E = 18. Using this value for E in the first two equations we created, we can also solve for P = 9 and M = 26. And as a check, we can put each of these values into the third equation: 18 + 26 + 9 = 53. Success!