Question

$$\begin{gathered} \frac{bx}{a}+\frac{ay}{b}=a^2+b^2, \\ x+y=2ab. \end{gathered}$$

Answer 1

The given equations are:

$\frac{bx}{a}+\frac{ay}{b}=a^2+b^2$
By taking LCM, we get:
$\frac{b^{2}x+a^{2}y}{ab}=a^2+b^2$
⇒ b²x + a²y = (ab)a² + b²
⇒ b²x + a²y = a³b + ab³ .......(i)

Also, x + y = 2ab........(ii)

On multiplying (ii) by a², we get:
a²x + a²y = 2a³b.........(iii)

On subtracting (iii) from (i), we get:
(b² − a²)x = a³b + ab³ − 2a³b
⇒ (b² − a²)x = −a³b + ab³
⇒ (b² − a²)x = ab(b² − a²)
⇒ (b² − a²)x = ab(b² − a²)
⇒ $x=\frac{ab\left(b^2-a^2\right)}{\left(b^2-a^2\right)}=ab$

On substituting x = ab in (i), we get:
b²(ab) + a²y = a³b + ab³
⇒ a²y = a³b
⇒ $\frac{a^{3}b}{a^2}=ab$
Hence, the solution is x = ab and y = ab.