Solve the following systems of simultaneous linear equations by the elimination method

px + qy = p - q

qx - py = p + q

Given,

px + qy = p - q …….(i)

qx - py = p + q …….(ii)

Multiplying eq. (i) by q and eq. (ii) by p we get,

pqx + q²y = pq - q² …….(iii)

pqx - p²y = p² + pq …….(iv)

Subtracting eq. (iv) from (iii) we get,

⇒ pqx + q²y - (pqx - p²y) = pq - q² - (p² + pq)

⇒ pqx - pqx + q²y + p²y = pq - pq - q² - p²

⇒ q²y + p²y = -q² - p²

⇒ y(q² + p²) = -(q² + p²)

⇒ y = $\dfrac{-(\text{q}^2 + \text{p}^2)}{\text{q}^2 + \text{p}^2}$ = -1.

Substituting value of y in eq. (i) we get,

⇒ px + q(-1) = p - q

⇒ px - q = p - q

⇒ px = p - q + q

⇒ px = p

⇒ x = 1.

Hence, x = 1 and y = -1.