Solve the following pair of equations: $(a \neq 0, b \neq 0)$
$\frac{2}{a} + \frac{3}{b} = 1$
$\frac{1}{a} + \frac{1}{b} = 1$

Form the pair of linear equations for the following problem,and find their solutions (if they exist) by the elimination method:
If we add $1$ to the numerator and subtract $1$ from the denominator,a fraction reduces to $1$. It becomes $\frac{1}{2}$ if we only add $1$ to the denominator. What is the fraction?

Solve the following pairs of equations by the method of elimination:
$5(x-2) + 3y = 1$
$3(x+1) + 5(y-3) = 1$

Solve the following pairs of equations by reducing them to a pair of linear equations:
$\frac{10}{x+y} + \frac{2}{x-y} = 4$
$\frac{15}{x+y} - \frac{5}{x-y} = -2$

Use the elimination method to find all possible solutions of the following pair of linear equations:
$2x + 3y = 8$ $...(1)$
$4x + 6y = 7$ $...(2)$

Solve the following pairs of equations:
$\frac{1}{2x} - \frac{1}{y} = -1$
$\frac{1}{x} + \frac{1}{2y} = 8, \quad x, y \neq 0$