Solving a Linear Equation System to Determine the Cost of 12 Shirts

Linear equations are fundamental to solving many real-world problems, including pricing and cost calculations. This article discusses a scenario involving sarees and shirts, which demonstrates how to solve a system of linear equations to find the cost of a specific number of items. We will walk through the process step-by-step and verify the accuracy of the solution.

Problem Statement

Given the following scenario:

The price of 2 sarees and 4 shirts is Rs. 1200. With the same amount of money, one can buy 1 saree and 6 shirts. We need to find the cost of buying 12 shirts.

Setting Up the Equations

To solve this problem, we will define the variables first:

x: price of the saree (in Rs) y: price of the shirt (in Rs)

Based on the given information, we can set up the following system of linear equations:

Equation 1:

2x 4y 1200

Equation 2:

x 6y 1200

Solving the System of Equations

Step 1: Solve one of the equations for one of the variables. Let's solve Equation 2 for x:

x 1200 - 6y

Step 2: Substitute this expression for x into Equation 1:

2(1200 - 6y) 4y 1200

Step 3: Simplify and solve for y:

2400 - 12y 4y 1200

-8y -1200

y 150

Step 4: Substitute the value of y back into the expression for x:

x 1200 - 6(150)

x 400

So, we have determined that:

The price of one saree (x) is 400 Rs. The price of one shirt (y) is 100 Rs.

Calculating the Cost of 12 Shirts

Now, we need to find the cost of buying 12 shirts:

Cost of 12 shirts 12 × 100 1200 Rs

Therefore, to buy 12 shirts, one will have to pay 1200 Rs.

Alternative Methods and Solutions

There are several alternative ways to solve the same problem. Here are a few examples:

Method 2: Direct Subtraction and Multiplication

Starting from the initial equations:

2x 4y 1200 x 6y 1200

We can subtract the second equation from the first equation after multiplying the second equation by 2:

2x 4y 1200 2x 12y 2400

Simplifying the subtraction:

4y - 12y 1200 - 2400

-8y -1200

y 150

Repeating the steps as described earlier:

Cost of 12 shirts 12 × 100 1200 Rs

Method 3: Direct Algebraic Manipulation

Akshay/Geekbot provided a direct method:

2x 4y 1200 — (1)

x 6y 1200 — (2)

Multiplying (2) by 2:

2x 12y 2400 — (3)

Subtract (1) from (3):

8y 1200

y 150

Repeating the steps as described earlier:

Cost of 12 shirts 12 × 100 1200 Rs

Method 4: Simplified Algebraic Solution

Jocker/Geekbot provided a simplified method:

2p 4s 1200 — (1)

p 6s 1200 — (2)

Multiplying (1) by 2:

4p 8s 2400 — (3)

(3) - (2):

3s 1200

s 200

Cost of 12 shirts 12 × 200 2400 Rs

Original misconception: The previous method was based on a misinterpreted initial condition. The correct cost for 12 shirts is 1200 Rs, not 2400 Rs as previously stated by Jocker/Geekbot.

Conclusion

By using a systematic approach and verifying the initial conditions, we can confidently determine the cost of buying 12 shirts as 1200 Rs. This problem reinforces the importance of careful algebraic manipulation and verification in solving linear equation systems.