Question

Solve for x and y:
$$\begin{gathered} \frac{x}{a}-\frac{y}{b}=0 \\ ax+by=a^2+b^2 \end{gathered}$$

Answer 1

The given equations are
$$\begin{gathered} \frac{x}{a}-\frac{y}{b}=0.....\left(i\right) \\ ax+by=a^2+b^2.....\left(ii\right) \end{gathered}$$
From (i)
$y=\frac{bx}{a}$
Substituting $y=\frac{bx}{a}$ in (ii), we get
$$\begin{gathered} ax+\frac{b\times bx}{a}=a^2+b^2 \\ \Rightarrow x=\frac{\left(a^2+b^2\right)\times a}{a^2+b^2}=a \end{gathered}$$
Now, substitute x = a in (ii) to get
$$\begin{gathered} a^2+by=a^2+b^2 \\ \Rightarrow by=b^2 \\ \Rightarrow y=b \end{gathered}$$
Hence, x = a and y = b.