Question

Solve for x and y:
$$\begin{gathered} px+qy=p-q \\ qx-py=p+q \end{gathered}$$

Answer 1

The given equations are
$$\begin{gathered} px+qy=p-q.....\left(i\right) \\ qx-py=p+q.....\left(ii\right) \end{gathered}$$
Multiplying (i) by p and (ii) by q and adding them, we get
$$\begin{gathered} p^{2}x+q^{2}x=p^2-pq+pq+q^2 \\ \Rightarrow x=\frac{p^2+q^2}{p^2+q^2}=1 \end{gathered}$$
Substituting x = 1 in (i), we have
$$\begin{gathered} p+qy=p-q \\ \Rightarrow qy=-q \\ \Rightarrow y=-1 \end{gathered}$$
Hence, x = 1 and $y=-1$.