Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method.
Yash scored $40$ marks in a test,getting $3$ marks for each right answer and losing $1$ mark for each wrong answer. Had $4$ marks 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?

(20) Let the number of right answers be $x$ and the number of wrong answers be $y$.
According to the given information:
For the first case: $3x - y = 40$ $...(1)$
For the second case: $4x - 2y = 50$
Dividing the second equation by $2$,we get: $2x - y = 25$ $...(2)$
Subtracting equation $(2)$ from equation $(1)$:
$(3x - y) - (2x - y) = 40 - 25$
$x = 15$
Substituting $x = 15$ in equation $(2)$:
$2(15) - y = 25$
$30 - y = 25$
$y = 5$
Therefore,the number of right answers is $15$ and the number of wrong answers is $5$.
Total number of questions $= x + y = 15 + 5 = 20$.