Question

Solve for x and y:

`x/a + y/b = 2, ax - by = (a^2 - b^2)`

Answer 1

The given equations are:

`x/a + y/b = 2`

⇒ `(bx + ay)/(ab) = 2`   ...[Taking LCM]

⇒ bx + ay = 2ab   ...(i)

Again, ax – by = (a² – b²)   ...(ii)

On multiplying (i) by b and (ii) by a, we get:

b²x + bay = 2ab²    ...(iii)

a²x – bay = a(a² – b²)   ...(iv)

On adding (iii) from (iv), we get:

(b² + a²)x = 2a²b + a(a² – b²)

⇒ (b² + a²)x = 2ab² + a³ – ab²

⇒ (b² + a²)x = ab² + a³

⇒ (b² + a²)x = a(b² + a²)

⇒ `x = (a(b^2 + a^2))/((b^2 + a^2)) = a`

On substituting x = a in (i), we get:

ba + ay = 2ab

⇒ ay = ab

⇒ y = b

Hence, the solution is x = a and y = b.