How to Solve Systems of Equations by Elimination

Oct 13, 2025 by miodeya.com 89 views

Introduction

Solving systems of equations is a fundamental concept in mathematics, and one of the most effective methods for solving these systems is by elimination. In this article, we will explore the concept of elimination and provide step-by-step solutions to three different systems of equations.

What is Elimination?

Elimination is a method of solving 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 the variable being eliminated.

How to Solve Systems of Equations by Elimination

To solve a system of equations by elimination, follow these steps:

Write down the equations: Write down the two equations in the system.
Identify the coefficients: Identify the coefficients of the variables in both equations.
Multiply the equations: Multiply one or both of the equations by a number that will make the coefficients of the variable being eliminated the same.
Add or subtract the equations: Add or subtract the equations to eliminate the variable.
Solve for the remaining variable: Solve for the remaining variable.
Substitute the value: Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

Example 1: Solving a System of Equations with Two Variables

System of Equations

{ \begin{align*} y - x &= 28 \\ y + x &= 156 \\ \end{align*} \}

Step 1: Write down the equations

The two equations in the system are:

y - x = 28

y + x = 156

Step 2: Identify the coefficients

The coefficients of the variables in both equations are:

y: 1, -1

x: -1, 1

Step 3: Multiply the equations

To make the coefficients of the variable being eliminated (x) the same, multiply the first equation by 1 and the second equation by 1.

Step 4: Add or subtract the equations

Add the two equations to eliminate the variable x.

(1)(y - x) + (1)(y + x) = (1)(28) + (1)(156)

y - x + y + x = 28 + 156

2y = 184

Step 5: Solve for the remaining variable

Divide both sides of the equation by 2 to solve for y.

y = 184/2

y = 92

Step 6: Substitute the value

Substitute the value of y into one of the original equations to solve for x.

y + x = 156

92 + x = 156

x = 156 - 92

x = 64

Solution

The solution to the system of equations is x = 64 and y = 92.

Example 2: Solving a System of Equations with Two Variables

System of Equations

{ \begin{align*} 3c + 6d &= 18 \\ 6c - 2d &= 22 \\ \end{align*} \}

Step 1: Write down the equations

The two equations in the system are:

3c + 6d = 18

6c - 2d = 22

Step 2: Identify the coefficients

The coefficients of the variables in both equations are:

c: 3, 6

d: 6, -2

Step 3: Multiply the equations

To make the coefficients of the variable being eliminated (d) the same, multiply the first equation by 1 and the second equation by 3.

Step 4: Add or subtract the equations

Add the two equations to eliminate the variable d.

(1)(3c + 6d) + (3)(6c - 2d) = (1)(18) + (3)(22)

3c + 6d + 18c - 6d = 18 + 66

21c = 84

Step 5: Solve for the remaining variable

Divide both sides of the equation by 21 to solve for c.

c = 84/21

c = 4

Step 6: Substitute the value

Substitute the value of c into one of the original equations to solve for d.

3c + 6d = 18

3(4) + 6d = 18

12 + 6d = 18

6d = 6

d = 1

Solution

The solution to the system of equations is c = 4 and d = 1.

Example 3: Solving a System of Equations with Two Variables

System of Equations

{ \begin{align*} 7x + 14y &= 28 \\ 5x + \end{align*} \}

Step 1: Write down the equations

The two equations in the system are:

7x + 14y = 28

5x + 10y = 20

Step 2: Identify the coefficients

The coefficients of the variables in both equations are:

x: 7, 5

y: 14, 10

Step 3: Multiply the equations

To make the coefficients of the variable being eliminated (x) the same, multiply the first equation by 5 and the second equation by 7.

Step 4: Add or subtract the equations

Subtract the second equation from the first equation to eliminate the variable x.

(5)(7x + 14y) - (7)(5x + 10y) = (5)(28) - (7)(20)

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

0 = 0

Step 5: Solve for the remaining variable

The equation 0 = 0 is a contradiction, which means that the system of equations has no solution.

Solution

The system of equations has no solution.

Conclusion

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

A: Elimination is a method of solving 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 the variable being eliminated.

Q: How do I know which variable to eliminate?

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

Q: What if the coefficients of the variable being eliminated are not the same?

A: If the coefficients of the variable being eliminated are not the same, you can multiply one or both of the equations by a number that will make the coefficients the same.

Q: Can I eliminate both variables at the same time?

A: No, you cannot eliminate both variables at the same time. Elimination involves adding or subtracting equations to eliminate one of the variables, leaving the other variable to be solved for.

Q: What if the resulting equation is a contradiction (e.g., 0 = 0)?

A: If the resulting equation is a contradiction, it means that the system of equations has no solution. This can occur when the equations are inconsistent or when the variables are not related to each other.

Q: Can I use elimination to solve systems of equations with more than two variables?

A: Yes, you can use elimination to solve systems of equations with more than two variables. However, you will need to eliminate one variable at a time, using the same steps as before.

Q: What are some common mistakes to avoid when using elimination?

A: Some common mistakes to avoid when using elimination include:

Not identifying the coefficients of the variables correctly
Not multiplying the equations by the correct numbers to make the coefficients the same
Not adding or subtracting the equations correctly
Not solving for the remaining variable correctly

Q: How can I practice using elimination to solve systems of equations?

A: You can practice using elimination to solve systems of equations by working through examples and exercises. Start with simple systems and gradually move on to more complex ones. You can also use online resources or math software to help you practice.

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

A: Solving systems of equations by elimination has many real-world applications, including:

Physics and engineering: Solving systems of equations is essential in physics and engineering to model real-world problems and make predictions.
Economics: Solving systems of equations is used in economics to model economic systems and make predictions about economic trends.
Computer science: Solving systems of equations is used in computer science to solve problems in computer graphics, game development, and other areas.

Conclusion

Solving systems of equations by elimination is a powerful method for solving these systems. By understanding the basics of elimination and practicing with examples and exercises, you can become proficient in solving systems of equations by elimination. Remember to identify the coefficients, multiply the equations, add or subtract the equations, solve for the remaining variable, and substitute the value. With practice, you will become proficient in solving systems of equations by elimination.