If (x + y) / z = 2, where x, y and z are real numbers with y ≠ z and x ≠ z, what is the value of the expression [y / (y - z)] + [x / (x - z)]?

Introduction / Context:
This question is an algebraic manipulation problem involving three variables x, y and z linked by the condition (x + y) / z = 2. You are asked to evaluate a rational expression involving y / (y - z) and x / (x - z) without solving for x and y individually. The problem tests the ability to use the given condition to substitute and simplify algebraic fractions efficiently.

Given Data / Assumptions:

• (x + y) / z = 2, so x + y = 2z.
• We must evaluate S = y / (y - z) + x / (x - z).
• Denominators y - z and x - z are non zero, so the expression is defined.
• x, y, z are real numbers.

Concept / Approach:
The idea is to use the condition x + y = 2z to express y in terms of x and z or vice versa, and then substitute into S. This reduces the number of variables and may cause cancellations. Alternatively, one can combine the two fractions in S into a single fraction and then use x + y = 2z to simplify the numerator. Both methods should lead to the same constant value for S, independent of the particular values of x, y and z that satisfy the condition.

Step-by-Step Solution:
Step 1: From (x + y) / z = 2, we have x + y = 2z. Step 2: Write the expression S as a single fraction. S = y / (y - z) + x / (x - z). Step 3: To make algebra easier, express y in terms of x and z. From x + y = 2z, we get y = 2z - x. Step 4: Substitute y = 2z - x into the first term. Then y - z = (2z - x) - z = z - x. So the first term becomes y / (y - z) = (2z - x) / (z - x). Step 5: Rewrite the second term x / (x - z). Notice that x - z = -(z - x), so x / (x - z) = x / (-(z - x)) = -x / (z - x). Step 6: Now S = (2z - x) / (z - x) - x / (z - x) = [(2z - x) - x] / (z - x). Step 7: Simplify the numerator: (2z - x - x) = 2z - 2x = 2(z - x). Step 8: Therefore S = 2(z - x) / (z - x) = 2, provided z ≠ x.

Verification / Alternative check:
Choose simple numbers that satisfy x + y = 2z. For example, let z = 5 and x = 3, then y = 2z - x = 10 - 3 = 7. Compute y / (y - z) = 7 / (7 - 5) = 7 / 2 and x / (x - z) = 3 / (3 - 5) = 3 / (-2) = -3 / 2. Their sum is 7 / 2 - 3 / 2 = 4 / 2 = 2. This matches the algebraic result. Any other consistent choice of x, y and z will yield the same value.

Why Other Options Are Wrong:
Options 0, 1, -1 and 1/2 represent other constant values that might arise from algebraic errors, such as incorrect handling of signs when using x - z = -(z - x) or from miscalculating the substitution y = 2z - x. Since we have shown that the expression always simplifies precisely to 2, none of these other values can be correct.

Common Pitfalls:
Students often forget that x - z and z - x are negatives of each other and fail to carry the minus sign through correctly. Another common issue is trying to guess numerical values without ensuring they satisfy the original condition x + y = 2z. Combining fractions without a common denominator or skipping steps in the simplification process can also lead to mistakes. Being systematic with substitutions and carefully handling signs prevents these problems.

Final Answer:
Thus, the value of the expression is 2.