Form the pair of linear equations in 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.

Let fixed charge be ₹ x and ₹ y be the additional charge per day.

Given,

Saritha paid ₹27 for a book kept for seven days.

For the first 3 days, she was charged the fix charge of ₹ x. For the remaining 4 days she was charged ₹ y per day.

∴ x + 4y = 27 ……..(1)

Susy paid ₹ 21 for a book for five days.

For the first 3 days, Susy was charged the fix charge of ₹ x. For the remaining 2 days she was charged ₹ y per day.

∴ x + 2y = 21 ………(2)

Subtracting equation (2) from (1), we get :

⇒ x + 4y - (x + 2y) = 27 - 21

⇒ x - x + 4y - 2y = 6

⇒ 2y = 6

⇒ y = $\dfrac{6}{2}$ = 3.

Substituting value of y in equation (2), we get :

⇒ x + 2(3) = 21

⇒ x + 6 = 21

⇒ x = 21 - 6

⇒ x = 15.

Hence, pair of linear equations are x + 4y = 27, x + 2y = 21, where x is the fixed charge (in ₹) and y is the additional charge (in ₹) per day; x = 15, y = 3.