Solving Simultaneous Equations to Determine the Cost of Pens and Pencils

When working with simultaneous equations, one can use various methods to find the values of the unknown variables. In this article, we will solve a set of equations to determine the cost of pens and pencils based on given conditions. This article will demonstrate the process step by step, and it will be beneficial for anyone interested in solving similar algebraic equations.

Problem Statement

The problem at hand is: If 2 pens and 3 pencils cost Rs 3500 and 3 pens and 2 pencils cost Rs 4000, what is the cost of a single pen?

Solution Using Algebraic Methods

Step 1: Define the Variables

Let's denote the cost of one pen as p and the cost of one pencil as q. We will then set up the following system of equations based on the given information.

Equation 1: 2p 3q 3500

Equation 2: 3p 2q 4000

Step 2: Eliminate One Variable

To solve these simultaneous equations, we will eliminate one of the variables. We can do this by manipulating the equations in a way that allows us to cancel out one of the terms.

First, multiply Equation 1 by 3 and Equation 2 by 2 to eliminate p: 3(2p 3q) 3(3500) which simplifies to 6p 9q 10500 (Equation 3) 2(3p 2q) 2(4000) which simplifies to 6p 4q 8000 (Equation 4)

Next, subtract Equation 4 from Equation 3 to eliminate p: 6p 9q - (6p 4q) 10500 - 8000 6q - 4q 2500

This simplifies to:

5q 2500 q 500

Step 3: Substitute Back to Find the Other Variable

Now that we have the value of q, we can substitute it back into Equation 1 to find p: 2p 3(500) 3500 2p 1500 3500 2p 3500 - 1500 2p 2000 p 1000

Conclusion

Therefore, the cost of one pen is Rs 1000.

Alternative Methods

Subtraction Method

Another method is to use the subtraction method to directly find the difference between the two equations:

Step 1: Take Equation 2 - Equation 1 3p 2q - (2p 3q) 4000 - 3500 3p - 2p 2q - 3q 500 p - q 500

Step 2: Now, multiply the new equation by 2 2p - 2q 1000

Step 3: Add this to Equation 1 2p - 2q 2p 3q 1000 3500 4p q 4500

Step 4: Subtract the modified Equation 2 4p q - (3p 2q) 4500 - 4000 4p q - 3p - 2q 500 p - q 500 q 500

Substituting q 500 into Equation 1, we find the value of p as 1000.

Final Answer: The cost of one pen is Rs 1000.

Conclusion

By solving simultaneous equations, we can accurately determine the individual costs of pens and pencils. This article demonstrates two effective methods to solve such problems. These techniques are not only useful in mathematical contexts but also in practical scenarios, such as inventory management and financial planning.

Key Takeaways

Understanding and applying simultaneous equations can help in solving real-world problems. Algebraic methods can be used to solve problems involving multiple unknowns. There are various methods (such as elimination, substitution, and addition-subtraction) to solve systems of equations.