Calculating Bus Ticket Fares: City A to B and C

This problem involves finding the cost of bus tickets between different cities based on given purchase scenarios. We can solve this using a system of linear equations.

Defining Variables for Fares

Let's define the unknown fares:

• Let the fare for a bus ticket from city A to city B be '$x$' (in Rs.).
• Let the fare for a bus ticket from city A to city C be '$y$' (in Rs.).

Formulating Equations from Given Information

We are given two conditions:

1. Two tickets from city A to B and three tickets from city A to C cost Rs. 77. This can be written as the equation:

   $2x + 3y = 77$ (Equation 1)

2. Three tickets from city A to B and two tickets from city A to C cost Rs. 73. This can be written as the equation:

   $3x + 2y = 73$ (Equation 2)

Solving the System of Linear Equations

We need to solve these two equations simultaneously to find the values of $x$ and $y$. We can use the elimination method.

Step 1: Multiply Equations to Match Coefficients

To eliminate one variable, we can multiply the equations by suitable numbers. Let's multiply Equation 1 by 3 and Equation 2 by 2, so the coefficients of '$x$' become the same.

Multiplying Equation 1 by 3:

$3 \times (2x + 3y) = 3 \times 77$
$6x + 9y = 231$ (Equation 3)

Multiplying Equation 2 by 2:

$2 \times (3x + 2y) = 2 \times 73$
$6x + 4y = 146$ (Equation 4)

Step 2: Eliminate One Variable

Now, subtract Equation 4 from Equation 3 to eliminate '$x$':

$(6x + 9y) - (6x + 4y) = 231 - 146$
$6x + 9y - 6x - 4y = 85$
$5y = 85$

Now, solve for '$y$':

$y = \frac{85}{5}$
$y = 17$

Step 3: Substitute to Find the Other Variable

Substitute the value of '$y$' (which is 17) back into either Equation 1 or Equation 2 to find '$x$'. Let's use Equation 1:

$2x + 3y = 77$
$2x + 3(17) = 77$
$2x + 51 = 77$

Now, solve for '$x$':

$2x = 77 - 51$
$2x = 26$
$x = \frac{26}{2}$
$x = 13$

Final Fares

So, the fare for a bus ticket from city A to B is Rs. 13, and the fare from city A to C is Rs. 17.

The fares for cities B and C from A are Rs. 13 and Rs. 17, respectively.