Question

Solve the following systems of equations:
$\frac{7x-2}{xy}=5$
$\frac{8x+7y}{xy}=15$

Answer 1

Given $\frac{7x-2y}{xy}=5\Rightarrow \frac{7x}{xy}-\frac{2y}{xy}=5\Rightarrow \frac{7}{y}-\frac{2}{x}=5\Rightarrow -\frac{2}{x}+\frac{7}{y}=5⟶\left(i\right)\frac{8x+7y}{xy}=15\Rightarrow \frac{8x}{xy}+\frac{7y}{xy}=15\Rightarrow \frac{8}{y}+\frac{7}{x}=15\Rightarrow \frac{7}{x}+\frac{8}{y}=15⟶\left(ii\right)$

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

$\therefore$ equation $\left(iii\right)$ and $\left(iv\right)$ simultaneously $,$

$-2u+7v=5\times 7\Rightarrow -14u+49v=35⟶\left(v\right)7u+8v=15\times 2\Rightarrow 14u+16v=30⟶\left(vi\right)$

Adding $\left(v\right)$ and $\left(vi\right),$

$65v=65$

$v=1$

putting value of $v$ in $\left(iii\right)$

$\therefore -2u+7v=5\Rightarrow -2u=5-7$$\Rightarrow u=1$

$\therefore u=1$ and $v=1$

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

$x=1$ and $y=1$