Question

Solve the pair of the equations
$\begin{array}{cc}& \frac{2}{x}+\frac{3}{y}=13\\ & \frac{5}{x}-\frac{4}{y}=-2\end{array}$

Answer 1

$\frac{2}{x}+\frac{3}{y}=13\frac{5}{x}-\frac{4}{y}=-2$

Let $\frac{1}{x}=u$ & $\frac{1}{y}=v$

Now the equations are ,

$2u+3v=13⟶\left(i\right)5u-4v=-2⟶\left(ii\right)$

Solving $\left(i\right)$ and $\left(Ii\right)$ ,

$2u+3v=13u=\frac{13-3v}{2}$

putting value of $u$ in $\left(ii\right)$ ,

$5u-4v=-25\left(\frac{13-3v}{2}\right)-4v=-2\Rightarrow -15v-8v=4-65\Rightarrow v=\frac{-69}{-23}=3\therefore v=3$

putting value of $v$ in $\left(i\right)$ equation ,

$2\cdot u+3\cdot 3=13u=2\therefore u=2$

& $v=3$

$\frac{1}{x}=u=2$ & $\frac{1}{y}=v=3$

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