First student bought art, math and history books for $72. a+m+h=72

Question

Second student bought art, chemistry and history books for $80. a+c+h=80
Third student bought chemistry, math and history books for $61. c+m+h=61
Fourth student bought art, math and chemistry books for $57. a+m+c=57
How much does each book cost?
How would you do this without guess and check?

Answer 1

You have 4 equations in 4 unknowns, so use elimination or substitution. For example,

a+0+h+m = 72
a+c+h+0 = 80
0+c+h+m = 61
a+c+0+m = 57

subtract #1 from #2 and #4
a+0+h+m = 72
0+c+0-m = 8
0+c+h+m = 61
0+c-h+0 = -15

subtract #2 from #3 and #4
a+0+h+m = 72
0+c+0-m = 8
0+0+h+2m = 53
0+0-h+m = -23

add #3 to #4
a+0+h+m = 72
0+c+0-m = 8
0+0+h+2m = 53
0+0+0+3m = 30

so, m=10
use that to fill in the other values, working your way up the rows.

Answer 2

subtracting 1st from 2nd ... c - m = 8
... c = m + 8

substitute into 3rd ... 2m + h = 53 (A)

substitute into 4th ... 2m + a = 49 (B)

subtracting ... h - a = 4

adding to 1st ... 2h + m = 76
... multiply by 2 ... 4h + 2m = 152
... subtract A ... 3h = 99 ... h = 33

substitute to find m , c , and a