Solve for variables from symmetric linear forms: Given (p/x) + (q/y) = m and (q/x) + (p/y) = n, determine x and y in terms of p, q, m, and n (assume denominators nonzero).

Introduction / Context:
This problem asks for an algebraic solution for x and y given two linear relations in the reciprocal variables 1/x and 1/y. The system is symmetric in p and q and can be solved elegantly by substituting u = 1/x and v = 1/y, turning the equations into a 2×2 linear system in u and v.

Given Data / Assumptions:

• (p/x) + (q/y) = m and (q/x) + (p/y) = n
• Assume x ≠ 0, y ≠ 0, and denominators in final expressions are nonzero (i.e., m p − n q ≠ 0 and n p − m q ≠ 0).

Concept / Approach:
Let u = 1/x and v = 1/y. Then p u + q v = m and q u + p v = n. Solve this linear system for u and v via determinants or elimination. Finally, take reciprocals to get x and y. Careful handling of the determinant p^2 − q^2 is essential.

Step-by-Step Solution:

Verification / Alternative check:
Substitute back: p/x = p * (m p − n q)/D and q/y = q * (n p − m q)/D; adding yields (m p^2 − n p q + n p q − m q^2)/D = m(p^2 − q^2)/D = m, as required. A similar check works for the second equation, confirming the formulae.

Why Other Options Are Wrong:
Options that swap p^2 − q^2 to q^2 − p^2 or flip denominator signs without matching both x and y expressions break the determinant structure and will not satisfy both original equations simultaneously.

Common Pitfalls:
Forgetting to convert to u and v, mishandling the determinant, or inverting signs in denominators leads to incorrect expressions. Always verify by substitution for symbolic solution questions.

Final Answer:
x = (p^2 − q^2)/(m p − n q), y = (p^2 − q^2)/(n p − m q)