Difficulty: Medium
Correct Answer: 1
Explanation:
Introduction / Context:
This algebra and geometry style problem uses a well known condition on three numbers: x^2 + y^2 + z^2 = xy + yz + zx. This condition appears often in coordinate geometry and vector problems and has an important interpretation. Here it is used to simplify an algebraic expression involving x, y, and z without explicitly solving for each variable separately.
Given Data / Assumptions:
Concept / Approach:
The key idea is to recognize what the condition x^2 + y^2 + z^2 = xy + yz + zx implies. If you bring all terms to one side, you can rewrite it as (x − y)^2 + (y − z)^2 + (z − x)^2 = 0. Since each square is always nonnegative for real numbers, the only way the sum of three squares can be zero is if each square itself is zero. This forces x, y, and z to be equal, which drastically simplifies the target expression.
Step-by-Step Solution:
Verification / Alternative check:
Pick any convenient nonzero value for k, for example k = 2. Then x = y = z = 2. The given condition is satisfied since x^2 + y^2 + z^2 = 4 + 4 + 4 = 12 and xy + yz + zx = 4 + 4 + 4 = 12. The expression becomes (14 + 6 − 10) / (10) = 10 / 10 = 1, which confirms the general reasoning.
Why Other Options Are Wrong:
The values 0, 5, 33/5, and 7 would arise only if x, y, and z were not all equal or if a mistake was made in simplifying the squared differences. Since the condition forces x, y, and z to be equal, the expression must simplify to a constant independent of their common value, and 1 is the only such constant that works.
Common Pitfalls:
Many learners do not recognize that x^2 + y^2 + z^2 = xy + yz + zx implies equality of x, y, and z, and they try to solve a complicated system instead. Others misexpand the squared terms when attempting to form (x − y)^2 and similar expressions. Practicing this identity helps you quickly simplify many higher level algebra and geometry problems.
Final Answer:
The value of the expression (7x + 3y − 5z) / (5x) is 1.
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