Solve the following pairs of equations by reducing them to a pair of linear equations:
$\frac{7x - 2y}{xy} = 5$
$\frac{8x + 7y}{xy} = 15$

(A) Given equations are:
$1$) $\frac{7x - 2y}{xy} = 5$
$2$) $\frac{8x + 7y}{xy} = 15$
Step $1$: Simplify the equations by dividing each term in the numerator by the denominator $xy$:
Equation $(1)$ becomes: $\frac{7x}{xy} - \frac{2y}{xy} = 5 \implies \frac{7}{y} - \frac{2}{x} = 5$
Equation $(2)$ becomes: $\frac{8x}{xy} + \frac{7y}{xy} = 15 \implies \frac{8}{y} + \frac{7}{x} = 15$
Step $2$: Let $u = \frac{1}{x}$ and $v = \frac{1}{y}$. The equations become:
$(i)$ $-2u + 7v = 5$
(ii) $7u + 8v = 15$
Step $3$: Solve the system of linear equations using the elimination method.
Multiply $(i)$ by $7$ and (ii) by $2$:
$-14u + 49v = 35$
$14u + 16v = 30$
Adding the two equations: $65v = 65 \implies v = 1$.
Step $4$: Substitute $v = 1$ into $(i)$:
$-2u + 7(1) = 5 \implies -2u = -2 \implies u = 1$.
Step $5$: Find $x$ and $y$:
Since $u = \frac{1}{x} = 1 \implies x = 1$.
Since $v = \frac{1}{y} = 1 \implies y = 1$.
Final Answer: $x = 1, y = 1$.