103 LearnersLast updated on July 4th, 2025
Elimination Method
To solve linear equations, we have different methods; one of the common methods used is the elimination method. It is applicable when there is a pair of equations. In this article, we will learn about the elimination method, its steps, and how to solve equations using it.
What is the Elimination Method?
The elimination method is a technique used to solve linear equations by eliminating a variable, either by multiplying or dividing both equations to make the coefficients of one variable equal. Then, using the basic arithmetic operations, we eliminate the variable, which is why the process is also known as the addition or addition-subtraction method.
What are the Steps to Use Elimination Method?
When solving linear equations using the elimination method, we remove one variable and then solve the equation using addition or subtraction. Follow these steps to use the elimination method:
Step 1: Write the equations in standard form
Before solving the equations, it is essential to check if the equations are in standard form or not, if not, arrange them in the form: ax + by = c.
Step 2: Multiplying or dividing the equation
First, we multiply or divide one or both equations by a non-zero number to make the coefficients of a variable equal.
Step 3: Adding or subtracting the equation to eliminate a variable
As the coefficients of a variable in both equations are the same, we can now add or subtract the equations to eliminate the variable.
Step 4: Simplifying the equation
Now we have an equation with one variable, and by solving the equation, we can find the value of this variable.
Step 5: Substitute to find the other variable
To find the value of the other variable, we substitute the value we found in the given equation and solve the equation.
For example, solving 2x + 3y = 12 and 4x - 3y = 6
Here, both equations are of the form ax + by = c and have one variable that has a common coefficient.
That is the coefficients of y are 3 and -3, as they have opposite signs, we add the equations.
(2x + 3y) + (4x - 3y) = 12 + 6
6x + 0y = 18
6x = 18
x = 18/6
x = 3
So, the value of x = 3
Find the value of y by substituting the value of x in 2x + 3y = 12
2(3) + 3y = 12
6 + 3y = 12
3y = 12 - 6
3y = 6
y = 6/3 = 2
Therefore, x = 3 and y = 2
Elimination Method: Infinitely Many Solutions
When the equations are coincident lines, which means both equations are on the same line. Then there will be more than one point of intersection, that is, every point on the line satisfies both equations. For these equations, if we use the elimination method, the answer will be 0 = 0. That means there are infinitely many solutions as x and y cancel out.
For example, solving x + 2y = 3 and 2x + 4y - 6 = 0
Arranging the equations in the form ax + by = c
x + 2y = 3 ⇒ equation 1
2x + 4y = 6 ⇒ equation 2
Multiplying equation 1 by 2:
(x + 2y) × 2 = 3 × 2
2x + 4y = 6
Subtract equation 1 from equation 2
(2x + 4y) - (2x + 4y) = 6 - 6
0 = 0
Solving System of 3 Equations Using Elimination Method
An equation with 3 variables is in the form Ax + By + Cz = D. We can use the elimination method to solve a system of 3 equations. We can learn it with an example.
Solve the system of equations: x + y + z = 6,
2x - y + z = 3,
3x + 2y - z = 4.
Here, the equations are in the standard form: Ax + By + Cz = D
x + y + z = 6 ⇒ 1
2x - y + z = 3 ⇒ 2
3x + 2y - z = 4 ⇒3
To eliminate z between equations 1 and 2, subtract equation 1 from 2:
(2x - y + z) - (x + y + z) = 3 - 6
x - 2y = -3
x - 2y = -3 ⇒ 4
Eliminating z in 1 and 3
Adding 1 and 3 to eliminate z
(x + y + z) + (3x + 2y - z) = 6 + 4
4x + 3y = 10 ⇒5
Now we have 2 equations with two variables
x - 2y = -3 ⇒ 4
4x + 3y = 10 ⇒ 5
Solving 4 and 5:
Subtracting 5 from 4
Multiplying equation 4 by 4: (x - 2y = -3) × 4
4x - 8y = -12
(4x - 8y) - (4x + 3y) = -12 - 10
0 - 11y = -22
y = -22/-11 = 2
Substituting y in equation 4:
x - 2(2) = -3
x - 4 = -3
x = -3 + 4
x = 1
Substituting x and y in equation 1
1 + 2 + z = 6
3 + z = 6
z = 6 - 3
z = 3
Here, x = 1, y = 2, z = 3
How to Solve Linear Equations Through Elimination Method?
The elimination method is used to solve two or more linear equations; it is not applicable to solve a single equation. The system of linear equations is a pair of linear equations. The system of linear equations is enclosed in the symbol ‘{’
To solve a linear equation using the elimination method, we eliminate the coefficients of one variable to find the value. If the equations have the same coefficient of one variable, we simply add or subtract the equations. If there are no same coefficients, we multiply or divide the equations and then add or subtract them.
Real-world applications of Elimination Method
In real life, we use the elimination method in fields like economics, budgeting, physics, and so on. Here are some real-world applications of the elimination method:
- To analyze the profit and cost, companies set up a system of equations, and by using the elimination method, we can determine the profit and cost.
- In chemistry, we use elimination methods to determine the proportions of different compounds needed to achieve the desired mixture.
- In physics, to solve the problems related to force, acceleration, and motion, we use the elimination method
Common Mistakes and How to Avoid Them in Elimination Method
When solving linear equations using the elimination method, errors are common among students, and they tend to repeat the same errors often. Here are some common mistakes and the ways to avoid them in the elimination method.
Mistake 1
Not aligning the terms
Students sometimes forget to align the terms in the standard form, and without proper order, it is hard to add or subtract from the equation. For example, if the equation is given in the form 2x - 3 = y, you first need to rearrange the equation in the form ax + by = C, that is, 2x - y = 3.
Mistake 2
Multiply only one side of the equation
When multiplying the equation to eliminate the value, students regularly multiply only one side of the equation. For instance, when multiplying 3x + 2y = 10 by 2, they multiply 3x + 2y by 2, but they won't multiply 10 by 2, which is incorrect. So when multiplying an equation by a number, we should multiply both sides of the equation, that is (3x + 2y = 10) × 2 = 6x + 4y = 20.
Mistake 3
Adding the equations when they should be subtracted
Confusing whether to add or subtract from the equation is common among students. For example, to eliminate the value of x in (x + y = 6) and (x - y = 2), students add the equations instead of subtracting, which is wrong. So always check whether to subtract or add the equation based on their signs.
Mistake 4
Errors while substitution
Substitution errors are common among students, that is, when y = 3 in x + 3y = 9, students sometimes substitute into the wrong equation. So when substituting, make sure you are placing the correct variable and always try to double-check the value.
Mistake 5
Multiplying the equation when it is not required
When there are common coefficients for a variable, we can directly add or subtract the equations. But students sometimes multiply or divide the equations even if there is a common coefficient, so always check if there are any common coefficients or not.
Solved Examples of Elimination Method
Problem 1
Solve x + y = 10 and x - y = 2
x = 6 and y = 4
Explanation
x + y =10
x - y = 2
To eliminate the y, add the equations:
(x + y) + (x - y) = 10 + 2
2x = 12
x = 12/2
x = 6
Substituting the value of x in x + y = 10
6 + y = 10
y = 10 - 6
y = 4
Problem 2
Solve 2x + 3y = 13 and 4x + 3y = 17
x = 2 and y = 3
Explanation
2x + 3y = 13
4x + 3y = 17
Subtracting the equations to eliminate y
(4x + 3y) - (2x + 3y) = 17 - 13
2x = 4
x = 4/2
x = 2
Substituting the value of x in 2x + 3y = 13 to find the value of y
2(2) + 3y = 13
4 + 3y = 13
3y = 13 - 4
3y = 9
y = 9/3
y = 3
Problem 3
Find two numbers whose sum is 9 and whose difference is 5
The two numbers are 7 and 2
Explanation
To find the numbers, let’s consider the numbers as x and y
x + y = 9 ⇒ 1
x - y = 5 ⇒ 2
Adding 1 and 2 to eliminate y
(x + y) + (x - y) = 9 + 5
2x = 14
x = 14/2
x = 7
Substituting the value of x in 1 to find y
x + y = 9 ⇒ 7 + y = 9
y = 9 - 7
y = 2
Problem 4
Find the value of m and n in 9m - 4n = 18 and 5m + 3n = 27?
Approximately, m = 3.44 and n = 3.24
Explanation
Let’s consider,
9m - 4n = 18 ⇒1
5m + 3n = 27 ⇒2
We multiply equations 1 by 3 and equation 2 by 4 to eliminate n
(9m - 4n = 18) × 3 ⇒ 27m - 12n = 54 ⇒3
(5m + 3n = 27) × 4 ⇒ 20m + 12n = 108 ⇒4
Adding 3 and 4
(27m - 12n) + (20m + 12n) = 54 + 108
47m = 162
m = 162 / 47
m ≈ 3.44
Substituting the value of m in 1
9m - 4n = 18
9 × 3.44 - 4n = 18
30.96 - 4n = 18
-4n = 18 - 30.96
-4n = -12.96
n = -12.6/-4
n = 3.24
Problem 5
The sum of two numbers is 30, and their difference is 10. Find the numbers.
Here, the numbers are 20 and 10
Explanation
Let the numbers be x and y
x + y = 30
x - y = 10
To eliminate the value of y, we add both equations
(x + y) + (x - y) = 30 + 10
2x = 40
x = 40/2
x = 20
Substituting the value of x in x + y = 30
20 + y = 30
y = 30 - 20
y = 10
FAQs on Elimination Method
1.What is the elimination method?
2.Can the elimination method be used to solve a linear equation with three variables?
3.What is a linear equation?
4.What are the different methods to solve linear equations?
5.What is the difference between the elimination and substitution methods?
6.How can children in Vietnam use numbers in everyday life to understand Elimination Method ?
7.What are some fun ways kids in Vietnam can practice Elimination Method with numbers?
8.What role do numbers and Elimination Method play in helping children in Vietnam develop problem-solving skills?
9.How can families in Vietnam create number-rich environments to improve Elimination Method skills?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
