If the real numbers x, y, and z satisfy xy + yz + zx = 1, what is the value of the expression (1 + y^2) / ((x + y)(y + z))?

Introduction / Context:
This question is a good example of algebraic simplification using given symmetric relations. You are told that xy + yz + zx = 1 and are asked to evaluate a rational expression involving x, y, and z. The key is to expand the denominator, recognise the given condition inside that expansion, and then simplify the expression to a single constant value.

Given Data / Assumptions:

• Real numbers x, y, and z satisfy xy + yz + zx = 1.
• The expression to evaluate is (1 + y^2) / ((x + y)(y + z)).
• All denominators are assumed non zero so the expression is well defined.
• We are looking for a simplified constant value, not a formula still involving x, y, and z.

Concept / Approach:
The strategy is to expand the denominator (x + y)(y + z) and see whether it contains the sum xy + yz + zx. Once we recognise that xy + yz + zx equals 1, we can substitute this value directly into the denominator. If the denominator simplifies to 1 + y^2, then the entire fraction becomes (1 + y^2) / (1 + y^2) = 1, provided this denominator is not zero. This type of manipulation is common in algebraic identity questions and trains you to spot known patterns inside larger expressions.

Step-by-Step Solution:
Start with the denominator: (x + y)(y + z).Expand it using distributive law: (x + y)(y + z) = xy + xz + y^2 + yz.Group terms as (xy + yz + xz) + y^2.From the given condition, xy + yz + zx = 1, which is the same as xy + yz + xz.Substitute this into the expansion: (x + y)(y + z) = 1 + y^2.Now the original expression becomes (1 + y^2) / (1 + y^2) = 1.

Verification / Alternative check:
We can test with a concrete example that satisfies xy + yz + zx = 1. Take x = 1, y = 0, and z = 1. Then xy + yz + zx = 1*0 + 0*1 + 1*1 = 1, so the condition is satisfied. Now evaluate the expression: numerator 1 + y^2 = 1 + 0^2 = 1. Denominator (x + y)(y + z) = (1 + 0)(0 + 1) = 1*1 = 1. Thus the value is 1/1 = 1, matching our algebraic simplification.

Why Other Options Are Wrong:

• Values 2, 3, 4, and 5 do not follow from the expansion; they would require a different relationship in the denominator.
• If the denominator had been something like 1 + 2y^2, the fraction might not simplify to 1, but in this case the given condition fits perfectly.
• Because the denominator exactly equals 1 + y^2, the ratio must be 1 whenever the expression is defined.

Common Pitfalls:

• Expanding (x + y)(y + z) incorrectly, for example missing one of the cross terms.
• Failing to recognise the grouped sum xy + yz + xz as the given value 1.
• Assuming the expression still depends on x, y, and z and trying to overcomplicate the answer instead of noticing the simple cancellation.

Final Answer:
1