Question

Solve the following simultaneous equations

$\frac{2}{x}-\frac{3}{y}=15;\frac{8}{x}+\frac{5}{y}=77$

Answer 1

$\frac{2}{x}-\frac{3}{y}=15$ .....$\left(1\right)$

$\frac{8}{x}+\frac{5}{y}=77$ .....$\left(2\right)$

Let $u=\frac{1}{x}$ and $v=\frac{1}{y}$ then eqns$\left(1\right)$ and $\left(2\right)$ becomes

$2u-3v=15$ .....$\left(3\right)$

$8u+5v=77$ .....$\left(4\right)$

$8\times$Eqn$\left(3\right)-2\times$Eqn$\left(4\right)$

$\Rightarrow 16u-24v-16u-10v=120-154$

$\Rightarrow -34v=-34$

$\Rightarrow v=\frac{-34}{-34}=1$

$\therefore v=1$

$\Rightarrow v=\frac{1}{y}=1$

$\therefore y=1$

Put $v=1$ in eqn$\left(3\right)$ we get

$2u-3\times 1=15$

$\Rightarrow 2u=15+3=18$

$\Rightarrow u=\frac{18}{2}=9$

$\therefore u=9$

$\Rightarrow u=\frac{1}{x}=9$

$\Rightarrow x=\frac{1}{9}$

Hence $x=\frac{1}{9}$ and $y=1$