If x + y = 2z, then the value of x/(x − z) + z/(y − z) is

Concept

When rational expressions are added, first write them over a common denominator and then simplify the resulting numerator and denominator.

Use the given algebraic relation at whichever stage most directly simplifies the expression, while preserving the restrictions that keep the original denominators nonzero.

Application

1. Let E = x/(x − z) + z/(y − z), with x + y = 2z.

2. Using the common denominator (x − z)(y − z), E = [x(y − z) + z(x − z)]/[(x − z)(y − z)].

3. Expand the numerator: xy − xz + xz − z2 = xy − z2.

4. Expand the denominator: (x − z)(y − z) = xy − xz − yz + z2 = xy − z(x + y) + z2.

5. Substitute x + y = 2z: xy − z(x + y) + z2 = xy − 2z2 + z2 = xy − z2.

6. Thus E = (xy − z2)/(xy − z2) = 1 wherever the original expression is defined.

Cross-check

Choose x = 3, y = 1, z = 2, which satisfies x + y = 2z. Then E = 3/(3 − 2) + 2/(1 − 2) = 3 − 2 = 1, confirming the result.