Question

Solve the following systems of equations:

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

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

Answer 1

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

Thus, the given equation reduces to

$2u+3v=13$...(i) and $5u-4y=-2$...(ii)

Multiplying (1) by $5$ and (ii) by $2$, we get,

$\Rightarrow 10u+15v=65$...(iii) and $10u-8y=-4$...(iv)

(iii) $-$ (iv), we get,

$\Rightarrow 23v=69$

$\therefore v=3$

Thus, $2u=13-3\left(3\right)=4$

$\therefore u=2$

Hence, $x=\frac{1}{u}=\frac{1}{2}$ and $y=\frac{1}{v}=\frac{1}{3}$