Solve the following pairs of equations by the method of elimination:
$7(y+3)-2(x+2)=14$
$4(y-2)+3(x-3)=2$

Solve the following pairs of linear equations: $\frac{2}{x} + \frac{3}{y} = \frac{9}{xy}, \quad \frac{4}{x} + \frac{9}{y} = \frac{21}{xy} \quad (x \neq 0, y \neq 0)$

Solve the following pairs of equations by the method of elimination:
$\frac{x}{2} + \frac{3y}{5} + 1 = 0$
$\frac{x}{2} + \frac{y}{3} - \frac{1}{3} = 0$

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

Solve the following pairs of equations by the method of elimination:
$9x + 7y = 55$
$7x + 9y = 57$

If the sum of two numbers is $20$ and their difference is $8,$ then the larger number is ...........