Question

Solve the following pair of linear equations:

$$\begin{gathered} px+qy=p-q \\ qx-py=p+q \end{gathered}$$

Answer 1

$$\begin{gathered} px+qy=p-q......\left(1\right) \\ qx-py=p+q......\left(2\right) \end{gathered}$$

Multiplying $p$ to equation $\left(1\right)$ and $q$ to equation $\left(2\right)$, we get

$$\begin{gathered} p^{2}x+pqy=p^2-pq\dots \dots \left(3\right) \\ {q}^{2}x-pqy=pq+q^2\dots \dots \left(4\right) \end{gathered}$$

Adding equation $\left(3\right)$ and equation $\left(4\right)$,we get

$$\begin{gathered} p^{2}x+q^{2}x=p^2+q^2 \\ \Rightarrow (p^2+q^2)x=p^2+q^2 \\ \Rightarrow x=\frac{(p^2+q^2)}{p^2+q^2} \\ \Rightarrow x=1 \end{gathered}$$

From equation $\left(1\right)$, we get

$$\begin{gathered} p\left(1\right)+qy=p–q \\ \Rightarrow qy=p-q-p \\ \Rightarrow qy=-q \\ \Rightarrow y=-1 \end{gathered}$$

Hence, $x=1$ and $\mathrm{y}=-1$ are the solution of the given pair of equations.