For the pair of equations $\frac{2}{x} + \frac{3}{y} = 3$ and $\frac{3}{x} + \frac{2}{y} = 7$,find the value of $\frac{3}{x} + \frac{3}{y}$.

For a pair of equations $13x - 7y = 19$ and $7x - 13y = 31$,find the value of $x + y$.

The angles of a cyclic quadrilateral $ABCD$ are $\angle A = (6x + 10)^{\circ}$,$\angle B = (5x)^{\circ}$,$\angle C = (x + y)^{\circ}$,and $\angle D = (3y - 10)^{\circ}$. Find $x$ and $y$,and hence the values of the four angles.

Places $A$ and $B$ are $90\, km$ apart from each other on a highway. $A$ car starts from $A$ and another from $B$ at the same time. If they move in the same direction,they meet in $9$ hours and if they move in opposite directions,they meet in $\frac{9}{7}$ hours. Find the speeds of the cars.

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

The students of a class are made to stand in rows. If $3$ students are extra in a row,there would be $1$ row less. If $3$ students are less in a row,there would be $2$ rows more. Find the number of students in the class.