Question

In any quarter, the college needs to make available 6 less English sections than Math sections.
Demand for Philosophy course is half as many sections as English sections.
Available classrooms limit the total sections of all three courses to 51.
Given these constraints how many sections of each course should the college make available each quarter?
So I am setting up a system of equations to solve for each variable. I use M for math courses, P for philosophy, E for English.
E=M-6 (we need 6 less English than Math)
P=2E (we need 2X English as Philosophy)
E+M+P=51

Answer 1

oops, so it should be E=2P because we need twice as many english classes as philosophy (or half as many philosophy as english)

Answer 2

Okay so I suppose I will answer my own question in case anyone else is curious.
So we have these three equations which we can solve by using substitution:

P=.5E
M=E+6
M+E+P=51

Sub in the first to equations for M & P so that we are working with only one variable.
E+6 (for M) .5E (for P) And we get this equation:
E+6+E+.5E=51
Combine like terms:
2.5E+6=51
isolate the variable by subtracting 6 from both sides:
2.5E=45
divide each side by 2.5:
E=18
plug that in to get our other variables.
M=E+6---->M=18+6--->M=24
P=.5E---> P=.5*18---> P=9
they need 24 math courses, 18 English courses, and 9 philosophy courses

Answer 3

the equations are correct.
I got the same solution.
to check your work plug your answers in the original.
good job!