Check whether the following pair of linear equation has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.

$3x-5y=20 , 6x-10y=40$

Does the following pair of linear equation have a unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.

$x-3y-7=0$ , $3x-3y-15=0$

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:

A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student $\mathrm{A}$ takes food for $20$ days she has to pay $\mathrm{₹}1000$ as hostel charges whereas a student $\mathrm{B}$, who takes food for $26$ days, pays $\mathrm{₹}1180$ as hostel charges. Find the fixed charges and the cost of food per day.

Find the pair of linear equations in the problem and find the solution (if it exists) 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.

Yash scored $40$ marks in a test, getting $3$ marks for each right answer and losing $1$ marks for each wrong answer. If $4$ marks had been awarded for each correct answer and $2$ marks been deducted for each incorrect answer, then Yash would have scored $50$ marks. How many questions were there in the test?

Form the pair of linear equation in the following problem and find its solution(if they exist) by any algebraic method:

Places $\mathrm{A}$ and $\mathrm{B}$ are $100\mathrm{}\mathrm{k}\mathrm{m}$ apart on a highway. One car starts from $\mathrm{A}$ and another from $\mathrm{B}$ at the same time. If the cars travel in the same direction at different speeds, they meet in $5$ hours. If they travel towards each other, they meet in $1$ hour. What are the speeds of the two cars?