Solve the following systems of equations:

The given equations are:

$\frac{4}{x}+3 y=14 \ldots$(i)

$\frac{3}{x}-4 y=23 \ldots(i i)$

Multiply equation $(i)$ by 4 and equation $(i i)$ by 3, add both equations, we get

$\frac{16}{x}+12 y=56$

Put the value of $x$ in equation $(i)$, we get

$\frac{4}{\frac{1}{5}}+3 y=14$

$\Rightarrow 3 y=-6$

$\Rightarrow y=-2$

Hence the value of $x=\frac{1}{5}$ and $y=-2$.