How to Use the Elimination Method

Oct 13, 2025 by miodeya.com 132 views

Introduction

Solving systems of 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 in detail, and provide step-by-step solutions to several systems of equations.

What is the Elimination Method?

The elimination method is a technique used to solve systems of equations by adding or subtracting equations to eliminate one of the variables. This method is based on the principle that if two equations are added or subtracted, the resulting equation will have the same coefficients for one of the variables, but the constant terms will be different.

How to Use the Elimination Method

To use the elimination method, follow these steps:

Write down the system of equations: Write down the system of equations that you want to solve.
Identify the coefficients: Identify the coefficients of the variables in each equation.
Choose a variable to eliminate: Choose a variable that has the same coefficient in both equations.
Add or subtract the equations: Add or subtract the equations to eliminate the chosen variable.
Solve for the remaining variable: Solve for the remaining variable using the resulting equation.
Substitute the value back into one of the original equations: Substitute the value of the remaining variable back into one of the original equations to solve for the other variable.

Example 1: Solving a System of Two Equations

Let's consider the following system of two equations:

y - x = 28

To solve this system, we can use the elimination method. We can add the two equations to eliminate the variable x.

y - x = 28 x + y = 28

Adding the two equations, we get:

2y = 56

Dividing both sides by 2, we get:

y = 28

Now that we have the value of y, we can substitute it back into one of the original equations to solve for x.

y - x = 28 28 - x = 28

Subtracting 28 from both sides, we get:

-x = 0

Multiplying both sides by -1, we get:

x = 0

Therefore, the solution to the system of equations is x = 0 and y = 28.

Example 2: Solving a System of Two Equations with Coefficients

Let's consider the following system of two equations:

3c + 6d = 18 6c - 2d = 22

To solve this system, we can use the elimination method. We can add the two equations to eliminate the variable c.

3c + 6d = 18 6c - 2d = 22

Adding the two equations, we get:

9c + 4d = 40

Now that we have the resulting equation, we can solve for c and d.

Multiplying the first equation by 3, we get:

9c + 18d = 54

Subtracting the second equation from the first equation, we get:

22d = 14

Dividing both sides by 22, we get:

d = 14/22

d = 7/11

Now that we have the value of d, we can substitute it back into one of the original equations to solve for c.

3c + 6d = 18 3c + 6(7/11) = 18

Multiplying both sides by 11, we get:

33c + 42 = 198

Subtracting 42 from both sides, we get:

33c = 156

Dividing both sides by 33, we get:

c = 156/33

c = 52/11

Therefore, the solution to the system of equations is c = 52/11 and d = 7/11.

Example 3: Solving a System of Two Equations with Coefficients

Let's consider the following system of two equations:

7x + 14y = 28 5x + 10y = 20

To solve this system, we can use the elimination method. We can add the two equations to eliminate the variable x.

7x + 14y = 28 5x + 10y = 20

Multiplying the first equation by 5 and the second equation by 7, we get:

35x + 70y = 140 35x + 70y = 140

Adding the two equations, we get:

70x + 140y = 280

Now that we have the resulting equation, we can solve for x and y.

Dividing both sides by 70, we get:

x + 2y = 4

Multiplying the first equation by 2, we get:

14x + 28y = 56

Subtracting the second equation from the first equation, we get:

-14x - 28y = -56

Multiplying both sides by -1, we get:

14x + 28y = 56

Now that we have the resulting equation, we can solve for x and y.

Dividing both sides by 14, we get:

x + 2y = 4

Subtracting 2y from both sides, we get:

x = 4 - 2y

Substituting this expression for x into one of the original equations, we get:

7(4 - 2y) + 14y = 28

Expanding and simplifying, we get:

28 - 14y + 14y = 28

Simplifying further, we get:

28 = 28

This is a true statement, so we can conclude that the solution to the system of equations is x = 4 - 2y.

Substituting this expression for x into one of the original equations, we get:

5(4 - 2y) + 10y = 20

Expanding and simplifying, we get:

20 - 10y + 10y = 20

Simplifying further, we get:

20 = 20

This is a true statement, so we can conclude that the solution to the system of equations is y = 4.

Therefore, the solution to the system of equations is x = 4 - 2y and y = 4.

Conclusion

In this article, we have explored the elimination method for solving systems of equations. We have provided step-by-step solutions to several systems of equations, and have shown how to use the elimination method to eliminate one of the variables. We have also shown how to solve for the remaining variable using the resulting equation. With practice and patience, you can become proficient in using the elimination method to solve systems of equations.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Glossary

System of Equations: A set of two or more equations that are related to each other.
Elimination Method: A technique used to solve systems of equations by adding or subtracting equations to eliminate one of the variables.
Coefficient: A number that is multiplied by a variable in an equation.
Variable: A letter or symbol that represents a value that can change.
Constant: A number that does not change value.
Equation: A statement that two expressions are equal.
Solution: A value or set of values that satisfy an equation or system of equations.
Frequently Asked Questions (FAQs) about Solving Systems of Equations by Elimination =====================================================================================

Q: What is the elimination method?

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

Q: How do I choose which variable to eliminate?

A: To choose which variable to eliminate, look for the coefficients of the variables in each equation. If the coefficients are the same, you can eliminate one of the variables by adding or subtracting the equations.

Q: What if the coefficients are not the same?

A: If the coefficients are not the same, you can multiply one or both of the equations by a number that will make the coefficients the same. This is called "multiplying by a scalar".

Q: How do I multiply an equation by a scalar?

A: To multiply an equation by a scalar, multiply each term in the equation by the scalar. For example, if you want to multiply the equation 2x + 3y = 5 by 2, you would get 4x + 6y = 10.

Q: What if I get a false statement when I eliminate a variable?

A: If you get a false statement when you eliminate a variable, it means that the system of equations has no solution. This is because the equations are inconsistent.

Q: How do I know if a system of equations has a solution?

A: To determine if a system of equations has a solution, you can use the following steps:

Write down the system of equations.
Eliminate one of the variables by adding or subtracting the equations.
Solve for the remaining variable.
Substitute the value of the remaining variable back into one of the original equations.
Check if the resulting equation is true.

Q: What if I get a true statement when I substitute the value of the remaining variable back into one of the original equations?

A: If you get a true statement when you substitute the value of the remaining variable back into one of the original equations, it means that the system of equations has a solution.

Q: How do I find the solution to a system of equations?

A: To find the solution to a system of equations, follow these steps:

Write down the system of equations.
Eliminate one of the variables by adding or subtracting the equations.
Solve for the remaining variable.
Substitute the value of the remaining variable back into one of the original equations.
Check if the resulting equation is true.

Q: What if I have a system of three or more equations?

A: If you have a system of three or more equations, you can use the elimination method to solve for one or more of the variables. You can also use other methods such as substitution or matrices to solve the system.

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

A: Yes, you can use the elimination method to solve systems of equations with fractions. To do this, multiply each term in the equation by the least common multiple (LCM) of the denominators.

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

A: Yes, you can use the elimination method to solve systems of equations with decimals. To do this, multiply each term in the equation by a power of 10 that will eliminate the decimals.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving systems of equations by elimination. We have covered topics such as choosing which variable to eliminate, multiplying an equation by a scalar, and determining if a system of equations has a solution. We have also provided step-by-step instructions on how to find the solution to a system of equations using the elimination method. With practice and patience, you can become proficient in using the elimination method to solve systems of equations.