(c) x = \(\frac{p^2-q^2}{mp-nq}\), y = \(\frac{p^2-q^2}{np-mq}\)
Let \(\frac{1}{X}\) = a and \(\frac{1}{y}\) = b. Then, the given equations reduce to
pa + qb = m .............(i)
qa + pb = n ..............(ii)
Multiplying eqn (i) by q and eqn (ii) by p, we get
pqa + q2b = mq …......(iii)
qpa + p2b = np .......…(iv)
Now, subtracting eqn (iii) from eqn (iv),
\((p^2-q^2)b\) = np-mq
\(\Rightarrow\)b = \(\frac{np-mq}{p^2-q^2}\) \(\Rightarrow\) y = \(\frac{1}{b}\) = \(\frac{p^2-q^2}{np-mq}\)
Substituting this value of b in (i), we have
pa + \(\frac{q(np-mq)}{p^2-q^2}\) = m
\(\Rightarrow\)pa = m - \(\frac{(pqn-mq^2)}{p^2-q^2}\)
\(\Rightarrow\)pa =
\(\Rightarrow\) pa = \(\frac{p(mp-qn)}{p^2-q^2}\)
\(\Rightarrow\) pa = \(\frac{mp-qn}{p^2-q^2}\)
\(\Rightarrow\) x = \(\frac{1}{a}\) = \(\frac{p^2-q^2}{mp-qn}\)