Solve:

ax + by = a - b
bx - ay = a + b

Given:

ax + by = a - b ……………….(1)
bx - ay = a + b ……………….(2)

Multiplying equation (1) by b:

(ax + by = a - b) x b

⇒ abx + b²y = ab - b² ……………….(3)

Multiplying equation (2) by a:

(bx - ay = a + b) x a

⇒ abx - a²y = a² + ab ……………….(4)

Subtract Equation (4) from Equation (3),

$\begin{matrix} & abx & + & b^2y & = & ab - b^2 \ & abx & - & a^2y & = & ab + a^2 \ & - & + & & & - \ \hline & & & (b^2 + a^2)y & = & ab - b^2 - (a^2 + ab) \ \Rightarrow & & & (b^2 + a^2)y & = & ab - b^2 - a^2 - ab \ \Rightarrow & & & (a^2 + b^2)y & = & - (a^2 + b^2) \ \Rightarrow & & & y & = & - \dfrac{(a^2 + b^2)}{(a^2 + b^2)} \ \end{matrix}$

⇒ y = -1

Substituting the value of y in equation (3), we get:

⇒ abx + b²(-1) = ab - b²

⇒ abx - b² = ab - b²

⇒ abx = ab

⇒ x = $\dfrac{ab}{ab}$ = 1

Hence, x = 1 and y = -1.