Solving Systems of Equations: Cost of Furniture Puzzle Solved

This article delves into solving a common linear algebra problem that arises in real-world scenarios, specifically in determining the cost of furniture. We will demonstrate how to solve the problem of finding the cost of 2 chairs and 2 tables given two sets of equations representing the total costs of different combinations of chairs and tables. Utilizing the principles of system of equations, we provide a detailed solution step by step, revealing the cost of the items involved and the techniques used in problem solving.

Problem Statement

The problem at hand is as follow: The total cost of 2 chairs and 3 tables is Rs. 800, while the total cost of 4 chairs and 3 tables is Rs. 1000. We need to determine the cost of 2 chairs and 2 tables.

Solution Method: System of Equations

To solve this problem, we utilize the method of system of equations. We start by representing the cost of one chair as x and the cost of one table as y. Thus, we can form the following equations from the problem statement:

Using Standard Algebraic Technique

Equation 1: (2x 3y 800)
Equation 2: (4x 3y 1000)

First, we solve for a single variable by eliminating one of them. To eliminate (y), we can multiply Equation 1 by 2 to align the coefficients of (x) in both equations: (2 times (2x 3y 800) Rightarrow 4x 6y 1600) Subtract Equation 2 from this result: ((4x 6y) - (4x 3y) 1600 - 1000 Rightarrow 3y 600) Solving for (y): (y frac{600}{3} 200) Now we substitute the value of (y) back into Equation 1 to solve for (x): (2x 3 times 200 800 Rightarrow 2x 600 800 Rightarrow 2x 200 Rightarrow x frac{200}{2} 100)

Alternative Method: Simplification by Combination

We can also simplify the solution by combining the given equations in a more straightforward manner:

Consider the equations: Equation 1: (2x 3y 800) Equation 2: (4x 3y 1000) By subtracting twice Equation 1 from Equation 2: ((4x 3y) - 2 times (2x 3y) 1000 - 2 times 800) This simplifies as: (4x 3y - 4x - 6y 1000 - 1600 Rightarrow -3y -600 Rightarrow y 200) Substituting (y 200) back into Equation 1: (2x 3 times 200 800 Rightarrow 2x 600 800 Rightarrow 2x 200 Rightarrow x frac{200}{2} 100)

Conclusion

Thus, the cost of one chair is Rs. 100 and the cost of one table is Rs. 200. To find the cost of 2 chairs and 2 tables, we perform the following calculation:

(2x 2y 2 times 100 2 times 200 200 400 600)

Therefore, the cost of 2 chairs and 2 tables is (boxed{Rs. 600}).

Conclusion

The system of equations method is a powerful tool for solving real-world problems related to determining the cost of items. Understanding and applying this method can help in various financial planning and inventory management tasks. While the problem involves basic algebra, the principles are widely applicable in more complex scenarios.