Given the equation (x - 5)^2 + (y - 2)^2 + (z - 9)^2 = 0 in three real variables x, y and z, what is the value of the linear expression x + y - z?

Introduction / Context:
This problem uses a simple but powerful idea from algebra and geometry. A sum of non negative square terms equals zero only when each individual square term is zero. The question then asks for a simple linear combination of x, y and z once you identify their exact values.

Given Data / Assumptions:
- (x - 5)^2 + (y - 2)^2 + (z - 9)^2 = 0.
- x, y and z are real numbers.
- We must find x + y - z.

Concept / Approach:
Each term (x - 5)^2, (y - 2)^2 and (z - 9)^2 is a square of a real number, hence each is greater than or equal to zero. A sum of such squares can be zero only if every term is individually zero. This observation quickly gives the values of x, y and z without any expansion.

Step-by-Step Solution:
Consider the expression S = (x - 5)^2 + (y - 2)^2 + (z - 9)^2. Each square term is non negative because a square of any real number is at least zero. The sum S is given to be zero. The only way a sum of non negative numbers can equal zero is if each number is itself zero. Therefore, (x - 5)^2 = 0, (y - 2)^2 = 0 and (z - 9)^2 = 0. Solving these, we get x - 5 = 0 so x = 5, y - 2 = 0 so y = 2, and z - 9 = 0 so z = 9. Now compute x + y - z = 5 + 2 - 9 = -2.

Verification / Alternative check:
Substituting x = 5, y = 2 and z = 9 back into the original expression gives (5 - 5)^2 + (2 - 2)^2 + (9 - 9)^2 = 0 + 0 + 0 = 0, which matches the condition. The computed value x + y - z = -2 is therefore consistent with the equation.

Why Other Options Are Wrong:
Values like 16, -1, 12 or 0 would require at least one of x, y or z to differ from 5, 2 or 9. In that case, at least one squared term would be positive and the sum could not be zero. So those options contradict the fundamental property of sums of squares.

Common Pitfalls:
A typical mistake is to expand all squares and attempt to solve a multivariable equation, which is unnecessary. Some learners also overlook that the condition forces a unique point in three dimensional space. Remember that a sum of squares equal to zero is a strong statement that each square is zero.

Final Answer:
The value of x + y - z is -2.