Solve for x and y: x a + y b =2, ax-by=a^2-b^2 - Vidyadip

Question

Solve for x and y:
$\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2,$
$\text{ax}-\text{by}=\text{a}^2-\text{b}^2$

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Answer

$\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$
$\frac{\text{bx}+\text{ay}}{\text{ab}}=2$
$bx + ay = 2ab ...(1)$
$ax - by = (a^2- b^2) ...(2)$
Multiplying $(1) $ by $b$ and $(2)$ by a
$\Rightarrow b^2x + bay = 2ab^2 ...(3)$
$\Rightarrow a^2x - bay = a(a^2 - b^2) ...(4)$
Adding $(3)$ and $(4),$
we get $b^2x + a^2x = 2ab^2 + a(a^2 - b^2) $
$x(b^2+ a^2) = 2ab^2 + a^3 - ab^2 $
$x(b^2 + a^2) = ab^2 + a^3 $
$x(b^2 + a^2) = a(b^2 + a^2)$
$\text{x}=\frac{\text{a}\big(\text{b}^2+\text{a}^2\big)}{\big(\text{b}^2+\text{a}^2\big)}=\text{a}$
Putting $x = a$ in $(1),$ we get $b \times a + ay= 2ab$
$ ay = 2ab - ab $
$ay = ab or y = b$
$\therefore$ Solution is $ x = a, y = b$

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