Question

Solve the following pairs of equations by reducing them to a pair of linear equations:
$\frac{1}{2x}+\frac{1}{3y}=2$ , $\frac{1}{3x}+\frac{1}{2y}=\frac{13}{6}$

Answer 1

Given equations are $\frac{1}{2x}+\frac{1}{3y}=2$ and $\frac{1}{3x}+\frac{1}{2y}=\frac{13}{6}$

Put $\frac{1}{x}=u,\frac{1}{y}=v$

$\Rightarrow \frac{1}{2}u+\frac{1}{3}v=2and\frac{1}{3}u+\frac{1}{2}v=\frac{13}{6}$

$\Rightarrow 3u+2v=12and2u+3v=13$

Solving the above equations we get $u=2,v=3$

Hence $\frac{1}{x}=u=2\Rightarrow x=\frac{1}{2}$ and $\frac{1}{y}=v=3\Rightarrow y=\frac{1}{3}$

Therefore $x=\frac{1}{2},y=\frac{1}{3}$