Solving a word problem using a system of linear equations of the form Ax + By = C

Question

Isabel the trainer has two solo workout plans that she offers her clients: Plan A and Plan B. Each client does either one or the other (not both). On Wednesday there were 5 clients who did Plan A and 3 who did Plan B. On Thursday there were 7 clients who did Plan A and 9 who did Plan B. Isabel trained her Wednesday clients for a total of 6 hours and her Thursday clients for a total of 12 hours.

How long does each of the workout plans last?

Workout plan A:

Workout plan B:

Answer 1

We have the two equations:

5A + 3B = 6

7A + 9B = 12

We can eliminate B by multiplying the first equation by 3:

15A + 9B = 18

7A + 9B = 12

Subtracting gives us

8A = 6

so plan A takes 6/8 of an hour, or 45 minutes.

Substituting that value for A in either of the original equations, we see that the time for plan B is also 45 minutes.