Check if the coefficients are opposites

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Ions by Elimination Practice IITo eliminate a variable, the coefficients must be opposites

Introduction

In the world of mathematics, particularly in algebra, elimination is a powerful technique used to solve systems of linear equations. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. In this article, we will delve into the concept of elimination and practice solving systems of linear equations using this technique.

What is Elimination?

Elimination is a method used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This technique is based on the principle that when two equations are added or subtracted, the coefficients of the variables must be opposites. In other words, if we want to eliminate a variable, the coefficients of that variable in both equations must be the same but with opposite signs.

How to Eliminate a Variable

To eliminate a variable, we need to follow these steps:

Identify the coefficients: Identify the coefficients of the variables in both equations.
Check if the coefficients are opposites: Check if the coefficients of the variable to be eliminated are opposites.
Add or subtract the equations: If the coefficients are opposites, add or subtract the equations to eliminate the variable.
Solve for the other variable: Once the variable is eliminated, solve for the other variable.

Practice Problem 1

Let's practice solving a system of linear equations using the elimination method. We have the following equations:

${$ -4x - 2y = -12$}$

${ $4x + 8y = -24\$}$

To eliminate the variable x, we need to add or subtract the equations. Since the coefficients of x in both equations are opposites, we can add the equations to eliminate x.

${$ (-4x - 2y) + (4x + 8y) = -12 + (-24)$}$

Simplifying the equation, we get:

${ $6y = -36\$}$

Now, we can solve for y by dividing both sides of the equation by 6:

${$ y = -6$}$

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

${$ -4x - 2(-6) = -12$}$

Simplifying the equation, we get:

${$ -4x + 12 = -12$}$

Subtracting 12 from both sides of the equation, we get:

${$ -4x = -24$}$

Dividing both sides of the equation by -4, we get:

${$ x = 6$}$

Therefore, the solution to the system of linear equations is x = 6 and y = -6.

Practice Problem 2

Let's practice solving another system of linear equations using the elimination method. We have the following equations:

${ $2x + 3y = 12\$}$

${$ -2x - 3y = -6$}$

To eliminate the variable x, we need to add or subtract the equations. Since the coefficients of x in both equations are opposites, we can add the equations to eliminate x.

${$ (2x + 3y) + (-2x - 3y) = 12 + (-6)$}$

Simplifying the equation, we get:

${ $0 = 6\$}$

This is a contradiction, which means that the system of linear equations has no solution.

Conclusion

In this article, we have learned how to solve systems of linear equations using the elimination method. We have practiced solving two systems of linear equations and have seen how to eliminate a variable by adding or subtracting equations. We have also seen how to solve for the other variable once the variable is eliminated. The elimination method is a powerful technique used to solve systems of linear equations, and it is an essential tool for any mathematician or scientist.

Frequently Asked Questions

What is elimination in mathematics?

Elimination is a method used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.
How do I eliminate a variable?

To eliminate a variable, you need to add or subtract the equations to eliminate the variable. The coefficients of the variable to be eliminated must be opposites.
What is the difference between elimination and substitution?

Elimination and substitution are two different methods used to solve systems of linear equations. Elimination involves adding or subtracting equations to eliminate a variable, while substitution involves solving one equation for one variable and substituting that value into the other equation.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Linear Algebra and Its Applications" by Gilbert Strang
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Introduction

In our previous article, we explored the concept of elimination and practiced solving systems of linear equations using this technique. In this article, we will answer some frequently asked questions about elimination and provide additional practice problems to help you master this method.

Q&A

Q: What is elimination in mathematics?

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

Q: How do I eliminate a variable?

A: To eliminate a variable, you need to add or subtract the equations to eliminate the variable. The coefficients of the variable to be eliminated must be opposites.

Q: What is the difference between elimination and substitution?

A: Elimination and substitution are two different methods used to solve systems of linear equations. Elimination involves adding or subtracting equations to eliminate a variable, while substitution involves solving one equation for one variable and substituting that value into the other equation.

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

A: Yes, you can use elimination to solve a system of linear equations with three variables. However, you will need to eliminate two variables at a time, and then use the resulting equation to solve for the third variable.

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

A: To determine which variable to eliminate first, you need to look at the coefficients of the variables in both equations. You should eliminate the variable with the smallest coefficient first.

Q: Can I use elimination to solve a system of linear equations with no solution?

A: Yes, you can use elimination to solve a system of linear equations with no solution. If the system has no solution, you will get a contradiction, such as 0 = 6.

Q: How do I know if a system of linear equations has no solution?

A: If the system of linear equations has no solution, you will get a contradiction, such as 0 = 6. This means that the system has no solution.

Practice Problems

Problem 1

Solve the following system of linear equations using elimination:

${ $2x + 3y = 12\$}$

${$ -2x - 3y = -6$}$

Problem 2

Solve the following system of linear equations using elimination:

${$ x + 2y = 6$}$

${$ -x - 2y = -3$}$

Problem 3

Solve the following system of linear equations using elimination:

${ $3x + 2y = 12\$}$

${$ -3x - 2y = -6$}$

Conclusion

In this article, we have answered some frequently asked questions about elimination and provided additional practice problems to help you master this method. We have also seen how to use elimination to solve systems of linear equations with three variables and how to determine which variable to eliminate first. With practice and patience, you will become proficient in using elimination to solve systems of linear equations.

Frequently Asked Questions

What is elimination in mathematics?

Elimination is a method used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.
How do I eliminate a variable?

To eliminate a variable, you need to add or subtract the equations to eliminate the variable. The coefficients of the variable to be eliminated must be opposites.
What is the difference between elimination and substitution?

Elimination and substitution are two different methods used to solve systems of linear equations. Elimination involves adding or subtracting equations to eliminate a variable, while substitution involves solving one equation for one variable and substituting that value into the other equation.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Linear Algebra and Its Applications" by Gilbert Strang
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton