Solve :

$\dfrac{9}{x} - \dfrac{4}{y} = 8$

$\dfrac{13}{x} + \dfrac{7}{y} = 101$

Given, equations :

$\Rightarrow \dfrac{9}{x} - \dfrac{4}{y} = 8$ …….(1)

$\Rightarrow \dfrac{13}{x} + \dfrac{7}{y} = 101$ …..(2)

Multiplying equation (1) by 7, we get :

$\Rightarrow 7\Big(\dfrac{9}{x} - \dfrac{4}{y}\Big) = 7 \times 8 \\[1em] \Rightarrow \dfrac{63}{x} - \dfrac{28}{y} = 56 \text{ ……(3)}$

Multiplying equation by (2) by 4, we get :

$\Rightarrow 4\Big(\dfrac{13}{x} + \dfrac{7}{y}\Big) = 4 \times 101 \\[1em] \Rightarrow \dfrac{52}{x} + \dfrac{28}{y} = 404 \text{ …….(4)}$

Adding equation (3) and (4), we get :

$\Rightarrow \Big(\dfrac{63}{x} - \dfrac{28}{y}\Big) + \Big(\dfrac{52}{x} + \dfrac{28}{y} \Big) = 56 + 404 \\[1em] \Rightarrow \dfrac{63 + 52}{x} = 460 \\[1em] \Rightarrow \dfrac{115}{x} = 460 \\[1em] \Rightarrow x = \dfrac{115}{460} = \dfrac{1}{4}.$

Substituting value of x in equation (1), we get :

$\Rightarrow \dfrac{9}{\dfrac{1}{4}} - \dfrac{4}{y} = 8 \\[1em] \Rightarrow 36 - \dfrac{4}{y} = 8 \\[1em] \Rightarrow \dfrac{4}{y} = 36 - 8 \\[1em] \Rightarrow \dfrac{4}{y} = 28 \\[1em] \Rightarrow y = \dfrac{4}{28} = \dfrac{1}{7}.$

Hence, x = $\dfrac{1}{4} \text{ and } y = \dfrac{1}{7}$.