Two bus tickets from city K to city L and three bus tickets from city K to city M cost Rs. 77 in total. Three tickets from city K to L and two tickets from city K to M cost Rs. 73 in total. What are the individual ticket fares from city K to L and from city K to M?

Introduction / Context:
This is a classic linear equations word problem involving two unknown fares. We know the total cost of different combinations of bus tickets between city K and two other cities L and M. Using these totals, we can set up a system of equations to find the individual ticket prices.

Given Data / Assumptions:
Let the fare from K to L be x rupees.Let the fare from K to M be y rupees.Two tickets K to L and three tickets K to M cost 77 rupees.Three tickets K to L and two tickets K to M cost 73 rupees.

Concept / Approach:
We translate the word statements into algebraic equations in variables x and y. There will be two linear equations, which we can solve using substitution or elimination. Once the values of x and y are found, we match them with the correct option that lists fares for L and M in the correct order.

Step-by-Step Solution:
Step 1: From the first statement, write 2x + 3y = 77.Step 2: From the second statement, write 3x + 2y = 73.Step 3: Use the elimination method. Multiply the first equation by 3: 6x + 9y = 231.Step 4: Multiply the second equation by 2: 6x + 4y = 146.Step 5: Subtract the second resulting equation from the first: (6x + 9y) − (6x + 4y) = 231 − 146.Step 6: This gives 5y = 85, so y = 17.Step 7: Substitute y = 17 into 3x + 2y = 73 to find x.Step 8: Compute 3x + 2 × 17 = 73, so 3x + 34 = 73 and 3x = 39, hence x = 13.

Verification / Alternative check:
Check the first condition with x = 13 and y = 17. Two tickets to L and three to M cost 2 × 13 + 3 × 17 = 26 + 51 = 77, matching the problem statement. Check the second condition: three tickets to L and two to M cost 3 × 13 + 2 × 17 = 39 + 34 = 73, also matching the given total. This confirms that the fares are correct.

Why Other Options Are Wrong:
The pair 17 and 19 would give totals 2 × 17 + 3 × 19 = 89 and 3 × 17 + 2 × 19 = 89, which do not match 77 and 73. The pairs 23 and 17, and 13 and 19 similarly fail when substituted into the given equations. Only 13 and 17 satisfy both cost equations simultaneously.

Common Pitfalls:
Errors often occur when setting up the equations, such as misplacing coefficients or mixing up which fare corresponds to which city. Another mistake is performing elimination incorrectly, leading to wrong values for x or y. Writing equations carefully and checking each substitution step helps avoid these issues.

Final Answer:
The fare from K to L is Rs. 13 and the fare from K to M is Rs. 17.