Introduction to the Elimination Method

Oct 13, 2025 by pinkandtrash.com 88 views

Introduction to the Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are multiples of each other. In this article, we will use the elimination method to solve a system of linear equations.

Understanding the System of Linear Equations

The system of linear equations we will be solving is given by:

$\begin{array}{c} -2x + 6y = -24 \\ 2x + 5y = -20 \end{array}$

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to make the coefficients of either $x$ or $y$ the same in both equations. We can do this by multiplying the equations by necessary multiples.

Let's multiply the first equation by 1 and the second equation by 1.

$\begin{array}{c} -2x + 6y = -24 \\ 2x + 5y = -20 \end{array}$

Step 2: Add or Subtract the Equations

Now that we have the equations with the same coefficients for one of the variables, we can add or subtract the equations to eliminate that variable.

Let's add the two equations to eliminate the variable $x$ .

$\begin{array}{c} -2x + 6y = -24 \\ 2x + 5y = -20 \end{array}$

Adding the two equations, we get:

$-2x + 2x + 6y + 5y = -24 - 20$

Simplifying the equation, we get:

$11y = -44$

Step 3: Solve for the Variable

Now that we have the equation with only one variable, we can solve for that variable.

Let's solve for $y$ .

$11y = -44$

Dividing both sides by 11, we get:

$y = -4$

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

Now that we have the value of one of the variables, we can substitute it into one of the original equations to solve for the other variable.

Let's substitute the value of $y$ into the first equation.

$-2x + 6(-4) = -24$

Simplifying the equation, we get:

$-2x - 24 = -24$

Step 5: Solve for the Other Variable

Now that we have the equation with only one variable, we can solve for that variable.

Let's solve for $x$ .

$-2x - 24 = -24$

Adding 24 to both sides, we get:

$-2x = 0$

Dividing both sides by -2, we get:

$x = 0$

Conclusion

Using the elimination method, we have solved the system of linear equations.

The values of $x$ and $y$ that satisfy both equations are:

$x = 0$ $y = -4$

Therefore, the correct solution is:

C. $(0, -4)$

Discussion

The elimination method is a powerful technique for solving systems of linear equations. By adding or subtracting the equations, we can eliminate one of the variables and solve for the other variable. This method is particularly useful when the coefficients of the variables in the two equations are multiples of each other.

In this article, we used the elimination method to solve a system of linear equations. We multiplied the equations by necessary multiples, added or subtracted the equations, solved for the variable, and substituted the value of the variable into one of the original equations to solve for the other variable.

The elimination method is a useful tool for solving systems of linear equations, and it is an important concept in mathematics and engineering.

Real-World Applications

The elimination method has many real-world applications, including:

Physics: The elimination method is used to solve systems of linear equations that describe the motion of objects in physics.
Engineering: The elimination method is used to solve systems of linear equations that describe the behavior of electrical circuits and mechanical systems.
Computer Science: The elimination method is used to solve systems of linear equations that arise in computer graphics and game development.

Conclusion

In conclusion, the elimination method is a powerful technique for solving systems of linear equations. By adding or subtracting the equations, we can eliminate one of the variables and solve for the other variable. This method is particularly useful when the coefficients of the variables in the two equations are multiples of each other.

The elimination method is a useful tool for solving systems of linear equations, and it is an important concept in mathematics and engineering.

References

"Linear Algebra and Its Applications" by Gilbert Strang
"Introduction to Linear Algebra" by Gilbert Strang
"Linear Algebra and Its Applications" by David C. Lay

Q: What is the elimination method?

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

Q: When can I use the elimination method?

A: You can use the elimination method when the coefficients of the variables in the two equations are multiples of each other.

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

A: To multiply the equations by necessary multiples, you need to find the least common multiple (LCM) of the coefficients of the variables in the two equations. Then, multiply each equation by the necessary multiple to make the coefficients of the variables the same.

Q: How do I add or subtract the equations?

A: To add or subtract the equations, you need to add or subtract the corresponding terms of the two equations. For example, if you have two equations:

$2x + 3y = 5$ $4x - 2y = 3$

You can add the two equations by adding the corresponding terms:

$(2x + 4x) + (3y - 2y) = 5 + 3$

Simplifying the equation, you get:

$6x + y = 8$

Q: How do I solve for the variable?

A: To solve for the variable, you need to isolate the variable on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same value.

Q: How do I substitute the value of the variable into one of the original equations?

A: To substitute the value of the variable into one of the original equations, you need to replace the variable with its value in the equation. For example, if you have an equation:

$2x + 3y = 5$

And you know that $x = 2$ , you can substitute the value of $x$ into the equation:

$2(2) + 3y = 5$

Simplifying the equation, you get:

$4 + 3y = 5$

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 the necessary multiples to make the coefficients of the variables the same.
Not adding or subtracting the equations correctly: Make sure to add or subtract the corresponding terms of the two equations.
Not solving for the variable correctly: Make sure to isolate the variable on one side of the equation.
Not substituting the value of the variable into one of the original equations correctly: Make sure to replace the variable with its value in the equation.

Q: What are some real-world applications of the elimination method?

A: The elimination method has many real-world applications, including:

Physics: The elimination method is used to solve systems of linear equations that describe the motion of objects in physics.
Engineering: The elimination method is used to solve systems of linear equations that describe the behavior of electrical circuits and mechanical systems.
Computer Science: The elimination method is used to solve systems of linear equations that arise in computer graphics and game development.

Q: How can I practice using the elimination method?

A: You can practice using the elimination method by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations using the elimination method on your own, and then check your answers with a calculator or online tool.

Q: What are some resources for learning more about the elimination method?

A: Some resources for learning more about the elimination method include:

Textbooks: "Linear Algebra and Its Applications" by Gilbert Strang, "Introduction to Linear Algebra" by Gilbert Strang, and "Linear Algebra and Its Applications" by David C. Lay.
Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram MathWorld.
Video tutorials: 3Blue1Brown, Crash Course, and Math Antics.