Q2(v):

Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (v) 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.

Given:

A lending library charges a fixed amount for the first three days and an additional charge for each day thereafter. Saritha paid ₹27 for a book kept for 7 days. Susy paid ₹21 for a book kept for 5 days.

To Find:

The fixed charge for the first three days and the additional charge for each extra day.

Step 1: Defining Variables

Let the fixed charge for the first three days be $x$ (in ₹).
Let the additional charge for each extra day be $y$ (in ₹).

Step 2: Formulating the Equations

For Saritha: She kept the book for 7 days. This includes 3 fixed days and 4 extra days ($7 - 3 = 4$).
The equation is: $x + 4y = 27$ --- (Equation 1)

For Susy: She kept the book for 5 days. This includes 3 fixed days and 2 extra days ($5 - 3 = 2$).
The equation is: $x + 2y = 21$ --- (Equation 2)

Step 3: Solving by Elimination Method

To eliminate $x$, we subtract Equation 2 from Equation 1:

$(x + 4y) - (x + 2y) = 27 - 21$

$x - x + 4y - 2y = 6$

$2y = 6$

$y = \frac{6}{2}$

$y = 3$

[Since the additional charge per day is ₹3]

Step 4: Finding the value of $x$

Substitute the value of $y = 3$ into Equation 2:

$x + 2(3) = 21$

$x + 6 = 21$

$x = 21 - 6$

$x = 15$

[Since the fixed charge for the first three days is ₹15]

Step 5: Verification

Check with Equation 1: $15 + 4(3) = 15 + 12 = 27$. (Matches the given condition for Saritha).

Final Answer: The fixed charge for the first three days is ₹15 and the charge for each extra day is ₹3.