Question

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

Answer 1

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