For a pair of equations $\frac{2}{x} + \frac{3}{y} = 17$ and $\frac{3}{x} + \frac{2}{y} = 13$,find the value of $\frac{1}{x} + \frac{1}{y}$.

Solve the following pairs of equations by the cross-multiplication method:
$4x - 19y + 13 = 0$
$13x - 23y + 19 = 0$

In a two-digit number,the digit at the unit's place is $(x+1)$ and the digit at the ten's place is $(x+2)$,then that number is $\ldots \ldots \ldots \ldots .$

For which values of $p$ and $q$ will the following pair of linear equations have infinitely many solutions?
$4x + 5y = 2$
$(2p + 7q)x + (p + 8q)y = 2q - p + 1$

For a pair of equations $9x - 4y = 14$ and $7x - 3y = 11$,find the value of $y$.

Graphically,solve the following pair of equations:
$2x + y = 6$
$2x - y + 2 = 0$
Find the ratio of the areas of the two triangles formed by the lines representing these equations with the $x$-axis and the lines with the $y$-axis.