The real variables x, y, z and the real constants p, q, r satisfy x/(pq - r^2) = y/(qr - p^2) = z/(pr - q^2). Given that...

Question

The real variables x, y, z and the real constants p, q, r satisfy x/(pq - r^2) = y/(qr - p^2) = z/(pr - q^2). Given that the denominators are non-zero, the value of px + qy + rz is

A. 0
B. 1
C. pq^r
D. p^2 + q^2 + r^2

Accepted Answer

px + qy + rz = 0 Three fractions share a common value k. This lets us write each of x, y, z as k times its denominator. Substituting into the target expression px + qy + rz, we find k times a bracket that turns out to be identically zero because every term in the expansion cancels with another term.

The real variables x, y, z and the real constants p, q, r satisfy x/(pq - r^2) = y/(qr - p^2) = z/(pr - q^2). Given that the denominators are non-zero, the value of px + qy + rz is A. 0 B. 1 C. pq^r D. p^2 + q^2 + r^2