How to Solve a System of Linear Equations by Elimination

Oct 13, 2025 by wbrum.org 93 views

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and one of the most effective methods for solving these systems is the elimination method. In this article, we will explore the elimination method and use it to solve a given system of linear equations.

What is the Elimination Method?

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the equations are relatively simple and can be easily manipulated.

How to Solve a System of Linear Equations by Elimination

To solve a system of linear equations by elimination, follow these steps:

Write down the system of linear equations: Write down the system of linear equations that you want to solve.
Identify the coefficients: Identify the coefficients of the variables in the equations.
Multiply the equations by necessary multiples: Multiply the equations by necessary multiples such that the coefficients of the variables to be eliminated are the same.
Add or subtract the equations: Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable: Solve for the remaining variable.
Back-substitute: Back-substitute the value of the remaining variable into one of the original equations to solve for the other variable.

Step-by-Step Solution to the Given System

Now, let's use the elimination method to solve the given system of linear equations.

Step 1: Write down the system of linear equations

The given system of linear equations is:

{ \begin{array}{l} -2x + 3y = 4 \\ -8x - 4y = 16 \end{array} \}

Step 2: Identify the coefficients

The coefficients of the variables in the equations are:

Coefficient of x in the first equation: -2
Coefficient of y in the first equation: 3
Coefficient of x in the second equation: -8
Coefficient of y in the second equation: -4

Step 3: Multiply the equations by necessary multiples

To eliminate the variable x, we need to multiply the first equation by 4 and the second equation by 1.

First equation multiplied by 4:

{ -8x + 12y = 16 }

Second equation remains the same:

{ -8x - 4y = 16 }

Step 4: Add or subtract the equations

Now, we can add the two equations to eliminate the variable x.

{ (-8x + 12y) + (-8x - 4y) = 16 + 16 }

Simplifying the equation:

{ -16x + 8y = 32 }

Step 5: Solve for the remaining variable

Now, we can solve for the variable y.

{ 8y = 32 + 16x }

{ y = \frac{32 + 16x}{8} }

{ y = 4 + 2x }

Step 6: Back-substitute

Now, we can back-substitute the value of y into one of the original equations to solve for the variable x.

Substituting y = 4 + 2x into the first equation:

{ -2x + 3(4 + 2x) = 4 }

Simplifying the equation:

{ -2x + 12 + 6x = 4 }

{ 4x = -8 }

{ x = -2 }

Step 7: Solve for the other variable

Now that we have the value of x, we can substitute it into the equation y = 4 + 2x to solve for the variable y.

{ y = 4 + 2(-2) }

{ y = 4 - 4 }

{ y = 0 }

Conclusion

In this article, we used the elimination method to solve a system of linear equations. We followed the steps of writing down the system of linear equations, identifying the coefficients, multiplying the equations by necessary multiples, adding or subtracting the equations, solving for the remaining variable, and back-substituting. We also solved for the other variable using the value of the remaining variable. The elimination method is a powerful tool for solving systems of linear equations, and it is particularly useful when the coefficients of the variables in the equations are relatively simple and can be easily manipulated.

Example Use Cases

The elimination method can be used to solve a wide range of systems of linear equations, including:

Simple systems: The elimination method can be used to solve simple systems of linear equations, such as the one given in this article.
Systems with multiple variables: The elimination method can be used to solve systems of linear equations with multiple variables.
Systems with complex coefficients: The elimination method can be used to solve systems of linear equations with complex coefficients.

Tips and Tricks

Here are some tips and tricks for using the elimination method:

Make sure to multiply the equations by necessary multiples: Make sure to multiply the equations by necessary multiples to eliminate one of the variables.
Add or subtract the equations carefully: Add or subtract the equations carefully to avoid making mistakes.
Solve for the remaining variable carefully: Solve for the remaining variable carefully to avoid making mistakes.
Back-substitute carefully: Back-substitute carefully to avoid making mistakes.

Conclusion

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.

Q: How do I know which variable to eliminate first?

A: To determine which variable to eliminate first, look for the coefficients of the variables in the equations. If the coefficients of the variables are the same, you can eliminate one of the variables by adding or subtracting the equations. If the coefficients of the variables are not the same, you may need to multiply the equations by necessary multiples to make the coefficients the same.

Q: How do I multiply the equations by necessary multiples?

A: To multiply the equations by necessary multiples, identify the coefficients of the variables in the equations and multiply the equations by the necessary multiples to make the coefficients the same.

Q: What is the difference between adding and subtracting equations?

A: When adding equations, you are combining the two equations to eliminate one of the variables. When subtracting equations, you are combining the two equations to eliminate one of the variables, but in the opposite direction.

Q: How do I solve for the remaining variable?

A: To solve for the remaining variable, use the equation that you obtained by adding or subtracting the equations. Solve for the remaining variable by isolating it on one side of the equation.

Q: What is back-substitution?

A: Back-substitution is the process of substituting the value of the remaining variable into one of the original equations to solve for the other variable.

Q: Why is it important to check the solution?

A: It is essential to check the solution to ensure that it satisfies both equations. If the solution does not satisfy both equations, it is not a valid solution.

Q: Can I use the elimination method to solve systems of linear equations with complex coefficients?

A: Yes, you can use the elimination method to solve systems of linear equations with complex coefficients. However, you may need to use more complex algebraic manipulations to eliminate the variables.

Q: Can I use the elimination method to solve systems of linear equations with multiple variables?

A: Yes, you can use the elimination method to solve systems of linear equations with multiple variables. However, you may need to use more complex algebraic manipulations to eliminate the variables.

Q: What are some common mistakes to avoid when using the elimination method?

A: Some common mistakes to avoid when using the elimination method include:

Not multiplying the equations by necessary multiples: Make sure to multiply the equations by necessary multiples to eliminate one of the variables.
Not adding or subtracting the equations carefully: Add or subtract the equations carefully to avoid making mistakes.
Not solving for the remaining variable carefully: Solve for the remaining variable carefully to avoid making mistakes.
Not back-substituting carefully: Back-substitute carefully to avoid making mistakes.

Q: How do I know if the elimination method is the best method to use?

A: The elimination method is a good choice when the coefficients of the variables in the equations are relatively simple and can be easily manipulated. However, if the coefficients of the variables are complex or the system of linear equations is large, you may want to consider using other methods, such as substitution or graphing.

Q: Can I use the elimination method to solve systems of linear equations with fractions?

A: Yes, you can use the elimination method to solve systems of linear equations with fractions. However, you may need to use more complex algebraic manipulations to eliminate the variables.

Conclusion

In conclusion, the elimination method is a powerful tool for solving systems of linear equations. By following the steps of writing down the system of linear equations, identifying the coefficients, multiplying the equations by necessary multiples, adding or subtracting the equations, solving for the remaining variable, and back-substituting, we can solve systems of linear equations using the elimination method. Remember to check the solution to ensure that it satisfies both equations and to avoid common mistakes when using the elimination method.