We can also rearrange the equation algebraically. If x = y + z, then:
y = x - z
z = x - y
Compare these to our answer choices:
(A) (y - z)/x
We know that y + z = x, so (y + z)/x = 1. If all terms are non-zero, then there's no way that y + z = y - z, so (y - z)/x must not equal 1.
(B) (y - x)/z
y - x is what we'd get if we multiplied (x - y) by -1. So we could say:
x - y = z --> y - x = -z
So (y - x)/z = -1, not 1.
(C) (z - x)/y
Similarly, z - x is what we'd get if we multiplied (x - z) by -1. So we could say:
x - z = y --> z - x = -y
So (z - x)/y = -1, not 1.
(D) (z - y)/x
We know that z + y = x, so (z + y)/x = 1. If all terms are non-zero, then there's no way that z + y = z - y, so (z - y)/x must not equal 1.
(E) (x - z)/y
Since we derived that x - z = y, we can substitute:
(x - z)/y = y/y = 1.
The answer is E.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education