Solving the System by Elimination

Oct 13, 2025 by miodeya.com 146 views

Introduction

In algebra, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of linear equations by elimination.

What is Elimination Method?

The elimination method is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of the variables in the two equations are additive inverses of each other. The elimination method involves the following steps:

Multiply the two equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
Add or subtract the two equations to eliminate the variable.
Solve the resulting equation for the remaining variable.
Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Solving the System by Elimination

Let's consider the following system of linear equations:

{ \begin{array}{l} -3x - 4y = -15 \\ 2x - 8y = -22 \end{array} \}

To solve this system by elimination, we need to eliminate one of the variables. Let's eliminate the variable x. We can do this by multiplying the first equation by 2 and the second equation by 3.

{ \begin{array}{l} -6x - 8y = -30 \\ 6x - 24y = -66 \end{array} \}

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

{ \begin{array}{l} -32y = -96 \end{array} \}

Solving for y, we get:

{ \begin{array}{l} y = 3 \end{array} \}

Now that we have the value of y, we can substitute it into one of the original equations to find the value of x. Let's substitute y = 3 into the first equation.

{ \begin{array}{l} -3x - 4(3) = -15 \end{array} \}

Simplifying the equation, we get:

{ \begin{array}{l} -3x - 12 = -15 \end{array} \}

Adding 12 to both sides of the equation, we get:

{ \begin{array}{l} -3x = -3 \end{array} \}

Dividing both sides of the equation by -3, we get:

{ \begin{array}{l} x = 1 \end{array} \}

Therefore, the solution to the system is x = 1 and y = 3.

Conclusion

In this article, we have learned how to solve a system of linear equations by elimination. We have seen how to eliminate one of the variables by adding or subtracting the equations, and how to solve the resulting equation for the remaining variable. We have also seen how to substitute the value of the remaining variable into one of the original equations to find the value of the other variable. With practice, you can become proficient in solving systems of linear equations by elimination.

Answer

The correct answer is A. (1, 3).

Discussion

This problem is a great example of how to solve a system of linear equations by elimination. The elimination method is a powerful tool for solving systems of linear equations, and it is essential to understand how to use it to solve problems like this one. If you have any questions or need further clarification, please don't hesitate to ask.

Practice Problems

Solve the system of linear equations by elimination:

{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}

Solve the system of linear equations by elimination:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 12 \end{array} \}

Solve the system of linear equations by elimination:

{ \begin{array}{l} x - 3y = -2 \\ 2x + 5y = 11 \end{array} \}

Glossary

Additive inverse: A number that is the opposite of another number.
Coefficient: A number that is multiplied by a variable.
Elimination method: A technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables.
Linear equation: An equation in which the highest power of the variable is 1.
System of linear equations: A set of two or more equations that involve two or more variables.
Solving a System of Linear Equations by Elimination: Q&A ===========================================================

Introduction

In our previous article, we learned how to solve a system of linear equations by elimination. In this article, we will answer some frequently asked questions about solving systems of linear equations by elimination.

Q: What is the elimination method?

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

Q: How do I know which variable to eliminate?

A: To determine which variable to eliminate, you need to look at the coefficients of the variables in the two equations. If the coefficients of the variables are additive inverses of each other, you can eliminate one of the variables by adding or subtracting the equations.

Q: What is an additive inverse?

A: An additive inverse is a number that is the opposite of another number. For example, the additive inverse of 3 is -3.

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

A: To multiply the equations by necessary multiples, you need to multiply both equations by the same number so that the coefficients of the variable to be eliminated are the same.

Q: What if I have a fraction in one of the equations?

A: If you have a fraction in one of the equations, you can multiply both equations by the denominator of the fraction to eliminate the fraction.

Q: Can I use the elimination method to solve a system of linear equations with three variables?

A: Yes, you can use the elimination method to solve a system of linear equations with three variables. However, you need to eliminate two variables at a time.

Q: What if I have a system of linear equations with no solution?

A: If you have a system of linear equations with no solution, it means that the equations are inconsistent. In this case, you can use the elimination method to find the inconsistency.

Q: Can I use the elimination method to solve a system of linear equations with infinitely many solutions?

A: Yes, you can use the elimination method to solve a system of linear equations with infinitely many solutions. However, you need to be careful when eliminating variables to avoid getting a false solution.

Q: How do I know if I have a system of linear equations with infinitely many solutions?

A: You can tell if you have a system of linear equations with infinitely many solutions by looking at the resulting equation after eliminating the variables. If the resulting equation is an identity (i.e., true for all values of the variable), then you have a system of linear equations with infinitely many solutions.

Q: Can I use the elimination method to solve a system of linear equations with complex numbers?

A: Yes, you can use the elimination method to solve a system of linear equations with complex numbers. However, you need to be careful when working with complex numbers to avoid making mistakes.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations by elimination. We hope that this article has helped you to better understand the elimination method and how to use it to solve systems of linear equations.

Practice Problems

Solve the system of linear equations by elimination:

{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}

Solve the system of linear equations by elimination:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 12 \end{array} \}

Solve the system of linear equations by elimination:

{ \begin{array}{l} x - 3y = -2 \\ 2x + 5y = 11 \end{array} \}

Glossary

Additive inverse: A number that is the opposite of another number.
Coefficient: A number that is multiplied by a variable.
Elimination method: A technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables.
Linear equation: An equation in which the highest power of the variable is 1.
System of linear equations: A set of two or more equations that involve two or more variables.