Solution:

The given equations are
$\begin{array}{l}
\mathrm{x}+\mathrm{y}=\mathrm{a}+\mathrm{b}\dots \dots(i) \\
\mathrm{ax}-\mathrm{by}=\mathrm{a}^{2}-\mathrm{b}^{2}\dots \dots(ii)
\end{array}$
Multiplying equation(i) by $\mathrm{b}$ and adding it with equation(ii), we obtain:
$\begin{array}{l}
b x+a x=a b+b^{2}+a^{2}-b^{2} \\
\Rightarrow x=\frac{a b+a^{2}}{a+b}=a
\end{array}$
Substituting $\mathrm{x}=\mathrm{a}$ in equation(i), we have
$\begin{array}{l}
a+y=a+b \\
\Rightarrow y=b
\end{array}$
As a result, $\mathrm{x}=\mathrm{a}$ and $\mathrm{y}=\mathrm{b}$.