Question

Solve the following simultaneous equations:
$\frac{1}{x}+\frac{1}{y}=8;$
$\frac{4}{x}-\frac{2}{y}=2$

Answer 1

Given, $\frac{1}{x}+\frac{1}{y}=8$
$\frac{4}{x}-\frac{2}{y}=2$
Substituting $\frac{1}{x}=a$ and $\frac{1}{y}=b$
$a+b=\mathrm{8..........}\left(1\right)$
$4a-2b=\mathrm{2.......}\left(2\right)$

Multiplying equations $\left(1\right)$ by $2$, we get

$2a+2b=\mathrm{16........}\left(3\right)$

Adding equation $\left(3\right)$ and equation $\left(2\right)$, we get

$2a+2b=16$
$4a-2b=2$
$6a=18$
$a=\frac{18}{6}$
$=3$

Substituting $a=3$ in equation $\left(1\right)$
$a+b=8$
$3+b=8$
$b=8-3$
$=5$
Resubstituting
$\frac{1}{x}=a$, $\frac{1}{y}=b$
$\frac{1}{x}=3$, $\frac{1}{y}=5$
$1=3x,1=5y$
$\therefore$ $x=\frac{1}{3}$, $y=\frac{1}{5}$