Q3(iv):

Form the pair of linear equations for the following problems and find their solution by substitution method. (iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is ` 105 and for a journey of 15 km, the charge paid is ` 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

Given:

1. The taxi charges consist of a fixed charge and a charge per kilometer.

2. For a distance of $10\text{ km}$, the total charge is $₹105$.

3. For a distance of $15\text{ km}$, the total charge is $₹155$.

To Find:

1. The fixed charge ($x$).

2. The charge per kilometer ($y$).

3. The total charge for a distance of $25\text{ km}$.

Step 1: Formulating the Linear Equations

Let the fixed charge be $x$ (in ₹) and the charge per kilometer be $y$ (in ₹/km).

Based on the problem statement, the total charge is given by the formula: $\text{Total Charge} = \text{Fixed Charge} + (\text{Distance} \times \text{Charge per km})$.

For the first condition ($10\text{ km}$ for $₹105$):

$x + 10y = 105$ --- (Equation 1)

For the second condition ($15\text{ km}$ for $₹155$):

$x + 15y = 155$ --- (Equation 2)

Step 2: Solving by Substitution Method

From Equation 1, express $x$ in terms of $y$:

$x = 105 - 10y$ --- (Equation 3)

Substitute the value of $x$ from Equation 3 into Equation 2:

$(105 - 10y) + 15y = 155$ [Substituting $x$]

$105 + 5y = 155$ [Combining like terms: $-10y + 15y = 5y$]

$5y = 155 - 105$ [Subtracting 105 from both sides]

$5y = 50$

$y = \frac{50}{5}$ [Dividing both sides by 5]

$y = 10$

Now, substitute $y = 10$ back into Equation 3 to find $x$:

$x = 105 - 10(10)$

$x = 105 - 100$

$x = 5$

Step 3: Calculating the Charge for 25 km

The total charge for $25\text{ km}$ is given by the expression: $x + 25y$.

Substitute $x = 5$ and $y = 10$ into the expression:

$\text{Total Charge} = 5 + 25(10)$

$\text{Total Charge} = 5 + 250$

$\text{Total Charge} = 255$

Final Answer:

The fixed charge is ₹5, the charge per km is ₹10, and the total charge for travelling 25 km is ₹255.