Solve the following pair of equations by the method of cross-multiplication:
$2x + y = 35$
$3x + 4y = 65$

Rajvansh invested some amount at the rate of $12 \%$ simple interest and some amount at the rate of $10 \%$ simple interest. He received a yearly interest of Rs. $130$. However,if he had interchanged the amounts invested,he would have received Rs. $4$ more as interest. How much amount did he invest at each rate?

There are some students in the two examination halls $A$ and $B$. To make the number of students equal in each hall,$10$ students are sent from $A$ to $B$. But if $20$ students are sent from $B$ to $A$,the number of students in $A$ becomes double the number of students in $B$. Find the number of students in the two halls.

The length of a rectangle is $5$ less than twice its breadth. If its perimeter is $110,$ find the area of the rectangle. Use the cross-multiplication method for the solution.

The sum of two positive numbers is $25$. Five times the smaller number is $5$ more than three times the larger number. Represent this as a pair of linear equations in two variables.

Solve the following pair of linear equations: $x+y=5xy$ and $3x+2y=13xy$.