Algebra Word Problem Setup

This problem involves finding the costs of individual items (bags and pens) based on the total cost of different combinations. We need to determine the cost of 9 bags and compare it to the cost of 2 pens.

Let's define the variables:

• Let \( b \) represent the cost of one bag in Rupees (₹).
• Let \( p \) represent the cost of one pen in Rupees (₹).

From the information given in the question, we can set up a system of two linear equations:

1. The cost of 18 bags and 3 pens is ₹1,164:

   \( 18b + 3p = 1164 \)

2. The cost of 14 bags and 2 pens is ₹922:

   \( 14b + 2p = 922 \)

Solving the System of Equations

To solve for \( b \) and \( p \), we can simplify the equations first. Divide Equation 1 by 3 and Equation 2 by 2:

• Simplified Equation 1: \( \frac{18b + 3p}{3} = \frac{1164}{3} \) which gives \( 6b + p = 388 \) (Equation 3)
• Simplified Equation 2: \( \frac{14b + 2p}{2} = \frac{922}{2} \) which gives \( 7b + p = 461 \) (Equation 4)

Now we can solve this simpler system. Let's use the elimination method. Subtract Equation 3 from Equation 4:

\( (7b + p) - (6b + p) = 461 - 388 \)

\( 7b + p - 6b - p = 73 \)

\( b = 73 \)

So, the cost of one bag ( \( b \) ) is ₹73.

Now, substitute the value of \( b \) back into Equation 3 to find \( p \):

\( 6b + p = 388 \)

\( 6(73) + p = 388 \)

\( 438 + p = 388 \)

\( p = 388 - 438 \)

\( p = -50 \)

The cost of one pen ( \( p \) ) is ₹-50. While a negative cost is unusual in a real-world scenario, we proceed with this value based on the given equations.

Calculating the Final Cost Difference

The question asks for the difference between the cost of 9 bags and the cost of 2 pens. This can be represented as \( 9b - 2p \).

Using the values we found for \( b \) and \( p \):

• Cost of 9 bags = \( 9 \times b = 9 \times 73 \)
• Cost of 2 pens = \( 2 \times p = 2 \times (-50) \)

Calculate the values:

• \( 9 \times 73 = 657 \)
• \( 2 \times (-50) = -100 \)

Now find the difference \( 9b - 2p \):

\( 9b - 2p = 657 - (-100) \)

\( 9b - 2p = 657 + 100 \)

\( 9b - 2p = 757 \)

The cost of 9 bags exceeds the cost of 2 pens by ₹757.