The solution of the equations (p/x) + (q/y) = m and (q/x) + (p/y) = n is

(p/x) + (q/y) = m ..(i)
(q/x) + (p/y) = n ...(ii)
On multiply Eq (i) by q and Eq. (ii) by p and subtracting, we get
(pq/x) + (q²)/y = mq
(pq/x) + (p²)/y = np
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(q²/y) - (p²/y) = mq - np
∴ (q² - p²) = y(mq - np)
∴ y = (q² - p²)/mp - np = (p² - q²)/np - mq
Again, on multiplying Eq. (i) by p and Eq. (ii) by q and subtracting, we get
(p²/x) + (pq/y) = mp
(q²/x) + (pq/y) = nq
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(p²/x) - (q²/x) = mp - nq
⇒ (p² - q²) = x (mp - nq)
⇒ x = (p² - q²)/(mp - nq)
∴ x = (p² - q²)/(mp - nq)
and y = (p² - q²) / (np - mq)