Solution:

The given equations are
$\begin{array}{l}
\mathrm{px}+\mathrm{qy}=\mathrm{p}-\mathrm{q}\dots \dots(i) \\
\mathrm{qx}-\mathrm{py}=\mathrm{p}+\mathrm{q}\dots \dots(ii)
\end{array}$
Multiplying equation(i) by $p$ and equation(ii) by $q$ and adding them, we obtain
$\begin{array}{l}
p^{2} x+q^{2} x=p^{2}-p q+p q+q^{2} \\
x=\frac{p^{2}+q^{2}}{p^{2}+q^{2}}=1
\end{array}$
Substituting $\mathrm{x}=1$ in equation(i), we have
$\begin{array}{l}
\mathrm{p}+\mathrm{qy}=\mathrm{p}-\mathrm{q} \\
\Rightarrow \mathrm{qy}=-\mathrm{p}
\end{array}$
$\Rightarrow \mathrm{y}=-1$
As a result, $\mathrm{x}=1$ and $\mathrm{y}=-1$