Question

$$\begin{gathered} \frac{x}{a}+\frac{y}{b}=2, \\ ax-by=\left(a^2-b^2\right). \end{gathered}$$

Answer 1

The given equations are:
$\frac{x}{a}+\frac{y}{b}=2$

⇒$\frac{bx+ay}{ab}=2$ [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)
⇒ $x=\frac{a\left(b^2+a^2\right)}{\left(b^2+a^2\right)}=a$
On substituting x = a in (i), we get:
ba + ay = 2ab
⇒ ay = ab
⇒ y = b

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