The cost of 8 pens and 10 pencils is ₹132. If the cost of a pen decreases by ₹2 and th cost of a pencil increases by ₹7, then the cost of 17 pens and 6 pencils is ₹136. What is the original cost of 12 pens and 9 pencils?

Correct Answer: ₹138 (Option 1)

Let's solve this step-by-step by setting up linear equations based on the information given in the question.

Step 1: Define the variables for original costs
Let the original cost of 1 pen be ₹$x$.
Let the original cost of 1 pencil be ₹$y$.

Step 2: Form the first equation
We are given that the total cost of 8 pens and 10 pencils is ₹132.
$8x+10y=132$

We can simplify this equation by dividing the entire equation by 2:
$4x+5y=66$ --- (Equation 1)

Step 3: Form the second equation based on altered costs
The cost of a pen decreases by ₹2, so the new cost of a pen is ₹$(x-2)$.
The cost of a pencil increases by ₹7, so the new cost of a pencil is ₹$(y+7)$.
With these new prices, the cost of 17 pens and 6 pencils is ₹136:
$17(x-2)+6(y+7)=136$

Expand and simplify this equation:
$17x-34+6y+42=136$
$17x+6y+8=136$
$17x+6y=136-8$
$17x+6y=128$ --- (Equation 2)

Step 4: Solve the system of linear equations
Multiply Equation 1 by 6 and Equation 2 by 5 to make the coefficients of $y$ equal:
$6\times (4x+5y)=6\times 66$ ⇒ $24x+30y=396$ --- (Equation 3)
$5\times (17x+6y)=5\times 128$ ⇒ $85x+30y=640$ --- (Equation 4)

Subtract Equation 3 from Equation 4:
$(85x+30y)-(24x+30y)=640-396$
$61x=244$
$x=\frac{244}{61}=4$

So, the original cost of 1 pen is ₹4.

Now, substitute $x=4$ into Equation 1 to find $y$:
$4\left(4\right)+5y=66$
$16+5y=66$
$5y=66-16$
$5y=50$
$y=10$

So, the original cost of 1 pencil is ₹10.

Step 5: Calculate the original cost of 12 pens and 9 pencils
$\text{Original Cost}=12x+9y$
$\text{Original Cost}=12\left(4\right)+9\left(10\right)$
$\text{Original Cost}=48+90=138$

Thus, the original cost of 12 pens and 9 pencils is ₹138.