Solving Linear Equations in Cost Determination: A System of Equations Approach

Solving systems of linear equations is a fundamental skill in mathematics, especially when dealing with real-world problems such as cost determination. This article will demonstrate how to use a system of equations to find the cost of individual items and determine the value of a variable based on total cost conditions. We will explore the step-by-step process using a practical example.

Introduction to the Problem

In this scenario, we are given the following information:

Two chairs and one table cost Rs. 700. One chair and two tables cost Rs. 800. The cost of m tables and m chairs is Rs. 30000.

Our goal is to determine the value of m.

Setting Up the Equations

Let's define the variables:

x - Let x be the cost of one chair. y - Let y be the cost of one table.

Based on the given information, we can write the following equations:

2x y 700 (Equation 1) x 2y 800 (Equation 2) mx my 30000 (Equation 3)

Solving the System of Equations

First, let's solve the first two equations to find the values of x and y.

Adding the Equations

By adding Equation 1 and Equation 2, we get:

(2x y) (x 2y) 700 800

3x 3y 1500

Dividing both sides by 3:

x y 500 (Equation 4)

Substituting into One of the Original Equations

Now, let's substitute y from Equation 1 into Equation 4:

2x (500 - x) 700

2x 500 - x 700

x 500 700

x 200

Now, substituting the value of x back into Equation 4:

200 y 500

y 300

So, the cost of one chair (x) is Rs. 200, and the cost of one table (y) is Rs. 300.

Solving for m

Now that we know the costs of chairs and tables, we can use this information to solve for m. The cost of m tables and m chairs is given as Rs. 30000.

From Equation 3, we have:

mx my 30000

Substituting the values of x and y:

m(200) m(300) 30000

500m 30000

m 60

Therefore, the value of m is 60.

Verifying the Solution

To verify, let's check if the cost of 60 tables and 60 chairs equals Rs. 30000:

60(200) 60(300) 12000 18000 30000

The solution is correct.

Conclusion

In conclusion, solving linear equations is a powerful tool for determining costs and solving real-world problems. By setting up a system of equations and solving it step-by-step, we can efficiently find the value of unknown variables.