Question

Solve each of the following pairs of equation by reducing then to a pair of linear equations.
$\frac{x+y}{xy}=2,\frac{x-y}{xy}=6$

Answer 1

$\frac{x+y}{xy}=2,\frac{x-y}{xy}=6$

$\Rightarrow \frac{x}{xy}+\frac{y}{xy}=2,\frac{x}{xy}-\frac{y}{xy}=6$

$\Rightarrow \frac{1}{y}+\frac{1}{x}=2,\frac{1}{y}-\frac{1}{x}=6$

Let $\frac{1}{x}=a$ and $\frac{1}{y}=b$, then

$\Rightarrow a+b=2...\left(1\right)$ and $-a+b=\mathrm{6...}\left(2\right)$

Equations (1)+(2), gives

$2b=8$

$\Rightarrow b=\frac{8}{2}=4$

Substitute $b=4$ in equation (1), gives

$a+4=2$

$\Rightarrow a=2-4=-2$

But $a=\frac{1}{x}$

$\Rightarrow -2=\frac{1}{x}$

$\Rightarrow x=-\frac{1}{2}$

$b=\frac{1}{y}$

$\Rightarrow 4=\frac{1}{y}$

$\Rightarrow y=\frac{1}{4}$

Thus, solution $(x,y)=(-\frac{1}{2},\frac{1}{4})$