Why Elimination is a Powerful Technique

Oct 13, 2025 by wbrum.org 110 views

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore the method of elimination, which is a powerful technique for solving systems of linear equations. We will use a specific example to illustrate the steps involved in solving a system of linear equations by elimination.

The Problem

If you multiply the first equation by 2, the following system is the result:

{ \begin{array}{l} 6x + 18y = -84 \\ -3x + 2y = -2 \end{array} \}

Our goal is to solve this system by elimination.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. In this case, we can multiply the first equation by 1 and the second equation by 9, so that the coefficients of x are 6 and -27, respectively.

{ \begin{array}{l} 6x + 18y = -84 \\ -27x + 18y = -18 \end{array} \}

Step 2: Add the Equations

Now that the coefficients of x are the same, we can add the equations to eliminate x.

{ \begin{array}{l} (6x - 27x) + (18y + 18y) = (-84) + (-18) \\ -21x + 36y = -102 \end{array} \}

Step 3: Solve for y

Now that we have eliminated x, we can solve for y. We can do this by dividing both sides of the equation by 36.

{ \begin{array}{l} \frac{-21x}{36} + \frac{36y}{36} = \frac{-102}{36} \\ -\frac{7}{12}x + y = -\frac{17}{6} \end{array} \}

Step 4: Substitute the Value of y into One of the Original Equations

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

{ \begin{array}{l} 6x + 18y = -84 \\ 6x + 18(-\frac{17}{6}) = -84 \end{array} \}

Step 5: Solve for x

Now that we have the value of y, we can solve for x. We can do this by simplifying the equation.

{ \begin{array}{l} 6x - 51 = -84 \\ 6x = -33 \\ x = -\frac{33}{6} \\ x = -\frac{11}{2} \end{array} \}

Conclusion

In this article, we have solved a system of linear equations by elimination. We multiplied the equations by necessary multiples, added the equations to eliminate x, solved for y, and substituted the value of y into one of the original equations to solve for x. The final answer is x = -\frac{11}{2} and y = -\frac{17}{6}.

Why Elimination is a Powerful Technique

Elimination is a powerful technique for solving systems of linear equations because it allows us to eliminate one of the variables by adding or subtracting the equations. This technique is particularly useful when the coefficients of the variables are the same or can be made the same by multiplying the equations by necessary multiples.

Real-World Applications of Elimination

Elimination has many real-world applications, including:

Physics and Engineering: Elimination is used to solve systems of linear equations that describe the motion of objects in physics and engineering.
Computer Science: Elimination is used to solve systems of linear equations that arise in computer science, such as in the solution of linear programming problems.
Economics: Elimination is used to solve systems of linear equations that arise in economics, such as in the solution of linear programming problems.

Tips and Tricks for Solving Systems of Linear Equations by Elimination

Here are some tips and tricks for solving systems of linear equations by elimination:

Make sure the coefficients of the variables are the same or can be made the same by multiplying the equations by necessary multiples.
Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.
Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

By following these tips and tricks, you can solve systems of linear equations by elimination with ease.

Conclusion

Q: What is elimination in the context of solving systems of linear equations?

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

Q: Why is elimination a powerful technique for solving systems of linear equations?

A: Elimination is a powerful technique for solving systems of linear equations because it allows us to eliminate one of the variables by adding or subtracting the equations. This technique is particularly useful when the coefficients of the variables are the same or can be made the same by multiplying the equations by necessary multiples.

Q: What are the steps involved in solving a system of linear equations by elimination?

A: The steps involved in solving a system of linear equations by elimination are:

Multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.
Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

Q: What are some common mistakes to avoid when solving systems of linear equations by elimination?

A: Some common mistakes to avoid when solving systems of linear equations by elimination include:

Not making sure the coefficients of the variables are the same or can be made the same by multiplying the equations by necessary multiples.
Not adding or subtracting the equations correctly to eliminate one of the variables.
Not solving for the remaining variable correctly by dividing both sides of the equation by the coefficient of the variable.
Not substituting the value of the remaining variable into one of the original equations to solve for the other variable correctly.

Q: What are some real-world applications of elimination in solving systems of linear equations?

A: Some real-world applications of elimination in solving systems of linear equations include:

Physics and Engineering: Elimination is used to solve systems of linear equations that describe the motion of objects in physics and engineering.
Computer Science: Elimination is used to solve systems of linear equations that arise in computer science, such as in the solution of linear programming problems.
Economics: Elimination is used to solve systems of linear equations that arise in economics, such as in the solution of linear programming problems.

Q: How can I practice solving systems of linear equations by elimination?

A: You can practice solving systems of linear equations by elimination by:

Solving problems from a textbook or online resource.
Creating your own problems and solving them.
Working with a partner or in a group to solve problems.

Q: What are some tips and tricks for solving systems of linear equations by elimination?

A: Some tips and tricks for solving systems of linear equations by elimination include:

Make sure the coefficients of the variables are the same or can be made the same by multiplying the equations by necessary multiples.
Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.
Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

By following these tips and tricks, you can solve systems of linear equations by elimination with ease.

Conclusion

In conclusion, elimination is a powerful technique for solving systems of linear equations. By multiplying the equations by necessary multiples, adding the equations to eliminate one of the variables, solving for the remaining variable, and substituting the value of the remaining variable into one of the original equations to solve for the other variable, we can solve systems of linear equations by elimination.