We have,

x + y = a + b …(i)

ax – by = a² – b²…(ii)

To solve these equations, we need to make one of the variables (in both the equations) have same coefficient.

Lets multiply equation (i) by b, so that variable y in both the equations have same coefficient.

Recalling equations (i) & (ii),

x + y = a + b [×b

ax – by = a² – b²

⇒ bx + ax = ab + a²

⇒ (b + a)x = a(b + a)

⇒ x = a

Substitute x = a in equations (i)/(ii), as per convenience of solving.

Thus, substituting in equation (i), we get

a + y = a + b

⇒ y = b

Hence, we have x = a and y = b.