PreCalculus

I need help with a Matrix Algebra problem

A small school has 100 students who occupy three classrooms A, B, and C. After the first period of the school day, half the students in room A move to room B, one-fifth of the students in room B move to room C, and one-third of the students in room C move to room A. Nevertheless, the total number of students in each room is the same for both periods. How many students occupy each room?

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  1. to start,
    a+b+c = 100

    After the room changes, we have

    a/2 + c/3 = a
    4b/5 + a/2 = b
    2c/3 + b/5 = c

    or,

    a/2 = c/3
    a/2 = b/5
    b/5 = c/3 = a/2

    so, substituting in,

    a + 5a/2 + 3a/2 = 100
    2a + 5a + 3a = 200
    a = 20
    b = 50
    c = 30
    during the move, 10 students move to each room.

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