How to Solve a Linear System Using the Elimination Method: Practical Guide

In this article, we will explain how to solve a linear system using the elimination method. We'll walk through a practical example and provide clear instructions to better illustrate the process. This will help you understand the method more clearly and apply it to similar problems.

Introduction to the Elimination Method

The elimination method is a technique used to solve a system of linear equations by eliminating one of the variables. The process involves adding or subtracting the equations in a way that one of the variables cancels out, allowing you to solve for the remaining variable.

The Problem

Let's consider the following linear system:

Equation A: 3x 2y 15

Equation B: 5x - 7y -73

Step-by-Step Solution

To solve this system using the elimination method, follow these steps:

Step 1: Align the Equations

Ensure that the variables are aligned:

3x 2y 15

5x - 7y -73

Step 2: Eliminate One Variable

Our goal is to eliminate one of the variables. To do this, we need to make the coefficients of one of the variables the same in both equations. Let's eliminate the variable y.

Multiply Equation A by 7 and Equation B by 2:

21x 14y 105

1 - 14y -146

Add the two equations to eliminate the variable y:

21x 14y 1 - 14y 105 - 146

31x -41

x frac{-41}{31}

Step 3: Substitute to Find the Other Variable

Now that we have the value of x, we can substitute it back into one of the original equations to find the value of y.

Substitute x frac{-41}{31} into Equation A:

3left(frac{-41}{31}right) 2y 15

frac{-123}{31} 2y 15

2y 15 frac{123}{31}

2y frac{465}{31} frac{123}{31}

2y frac{588}{31}

y frac{588}{62}

y frac{294}{31}

Converting the fraction to a decimal:

y approx 9.48

Conclusion

In this example, we solved the linear system using the elimination method. By eliminating one variable, we were able to find the values of both variables.

Understanding and mastering the elimination method is crucial for solving more complex systems of linear equations. It is also a valuable technique for practical applications in mathematics and real-world problems.

Frequently Asked Questions

Q: What is the elimination method?

A: The elimination method is a technique used to solve a system of linear equations by eliminating one of the variables through addition or subtraction of the equations.

Q: Can you use the elimination method with any system of linear equations?

A: Yes, the elimination method can be applied to any system of linear equations. However, it may require some manipulation of the equations to align the coefficients.

Q: What if the coefficients of the variables are not the same in the equations?

A: If the coefficients are not the same, you can multiply one or both equations by a constant so that one of the variables has the same coefficient in both equations, allowing you to eliminate that variable.

Related Articles

Explore more related articles and learn about other methods to solve linear systems:

How to Solve a Linear System Using the Substitution Method An Algebraic Approach to Solving Linear Systems Real-World Applications of Linear Systems in Mathematics