Question

$$\begin{gathered} \frac{bx}{a}-\frac{ay}{b}+a+b=0, \\ bx-ay+2ab=0. \end{gathered}$$

Answer 1

The given equations are:
$\frac{bx}{a}-\frac{ay}{b}+a+b=0$
By taking LCM, we get:
b²x − a²y = −a²b − b²a .......(i)

and bx − ay + 2ab = 0
bx − ay = −2ab ........(ii)

On multiplying (ii) by a, we get:
abx − a²y = −2a²b .......(iii)

On subtracting (i) from (iii), we get:
abx − b²x = − 2a²b + a²b + b²a = −a²b + b²a
⇒ x(ab − b²) = −ab(a − b)
⇒ x(a − b)b = −ab(a − b)
∴ $x=\frac{-ab\left(a-b\right)}{\left(a-b\right)b}=-a$

On substituting x = −a in (i), we get:
b²(−a) − a²y = −a²b − b²a
⇒ −b²a − a²y = −a²b − b²a
⇒ −a²y = −a²b
⇒ y = b

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