Form the pair of linear equations for the following problem,and find their solutions (if they exist) by the elimination method:
$A$ lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ $27$ for a book kept for seven days,while Susy paid ₹ $21$ for the book she kept for five days. Find the fixed charge and the charge for each extra day.

(N/A) Let the fixed charge for the first three days be $Rs$ $x$ and the additional charge for each day thereafter be $Rs$ $y$.
According to the given information:
For Saritha: The book was kept for $7$ days. This includes $3$ fixed days and $4$ extra days. So,$x + 4y = 27$ $...(1)$
For Susy: The book was kept for $5$ days. This includes $3$ fixed days and $2$ extra days. So,$x + 2y = 21$ $...(2)$
Subtracting equation $(2)$ from equation $(1)$:
$(x + 4y) - (x + 2y) = 27 - 21$
$2y = 6$
$y = 3$ $...(3)$
Substituting the value of $y = 3$ in equation $(1)$:
$x + 4(3) = 27$
$x + 12 = 27$
$x = 15$
Therefore,the fixed charge is $Rs$ $15$ and the charge for each extra day is $Rs$ $3$.