From the Dharmapuri bus stand if we buy 2 tickets to Palacode and 3 tickets to Karimangalam the total cost is Rs. 32, but if we buy 3 tickets to Palacode and one ticket to Karimangalam, the total cost is Rs. 27. Find the fares from Dharmapuri to Palacode and Karimangalam.

Hint: Assume the cost of tickets from Dharmapuri to Palacode and Dharmapuri to Karimangalam as variables and write the equations involving them. Then solve the two equations in two variables to find the fares from Dharmapuri to Palacode and Karimangalam.

Complete step-by-step answer:
Let the cost of a ticket from Dharmapuri to Palacode be Rs. x.
Let the cost of a ticket from Dharmapuri to Karimangalam be Rs. y.
It is given that if we buy two tickets to Palacode and three tickets to Karimangalam the total cost is Rs. 32. Hence, we have:
\[2x + 3y = 32............(1)\]
It is also given that if we buy three tickets for Palacode and one ticket to Karimangalam, then the total cost is Rs. 27. Hence, we have:
\[3x + y = 27............(2)\]
We now have two equations in two variables. We can solve them.
Multiply equation (1) by 3 and the equation (2) by 2 and subtract them.
\[3(2x + 3y) - 2(3x + y) = 3(32) - 2(27)\]
Simplifying, we get:
$\Rightarrow$ \[6x + 9y - 6x - 2y = 96 - 54\]
$\Rightarrow$ \[7y = 42\]
Solving for y, we have:
$\Rightarrow$ \[y = \dfrac{{42}}{7}\]
$\Rightarrow$ \[y = 6.............(3)\]
Substituting equation (3) in equation (2), we get:
$\Rightarrow$ \[3x + 6 = 27\]
Simplifying, we get:
$\Rightarrow$ \[3x = 27 - 6\]
$\Rightarrow$ \[3x = 21\]
Solving for x, we get:
$\Rightarrow$ \[x = \dfrac{{21}}{3}\]
$\Rightarrow$ \[x = 7\]
Hence, the value of fare from Dharmapuri to Palacode is Rs. 7 and the value of fare from Dharmapuri to Karimangalam is Rs.6.

Note: After finding the value of the variables, you can substitute them in the two equations to check if your answer is correct or not.