In a two-digit number,the digit at unit's place is $x$ and the digit at ten's place is $y$. The number obtained by interchanging the digits is ...........

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?

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method. $A$ fraction becomes $\frac{1}{3}$ when $1$ is subtracted from the numerator and it becomes $\frac{1}{4}$ when $8$ is added to its denominator. Find the fraction.

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

$A$ boat goes $32 \, km$ upstream and $36 \, km$ downstream in $7$ hours. In $9$ hours,it can go $40 \, km$ upstream and $48 \, km$ downstream. Determine the speed of the stream and that of the boat in still water. (Speed of boat in still water is more than the speed of the stream of river.)

For all real values of $c,$ the pair of equations $x-2y=8$ and $5x-10y=c$ have a unique solution. Justify whether it is true or false.