How do you solve a system of equations by elimination

The elimination method of solving a system of linear equations algebraically is the most widely used method out of all the methods to solve linear equations. In the elimination method, we eliminate any one of the variables by using basic arithmetic operations and then simplify the equation to find the value of the other variable. Then we can put that value in any of the equations to find the value of the variable eliminated.

The elimination method is easy to use because here we eliminate one of the terms that make our calculations simple. In this article, we will learn how to solve systems of equations using the elimination method. We shall solve various examples based on the concept for a better understanding. 

What is Elimination Method?

As per the elimination method definition, it is about eliminating one of the terms containing any of the variables to make the calculations easier. This is done by multiplying or dividing a number by the equation(s) such that the coefficients of any one of the variable terms become the same. Then, we add or subtract both the equations to eliminate or remove that term from the result. That is why the elimination method is also called the addition method. For example, let us solve two linear equations containing two variables using the elimination method.

Solving Systems of Equations by Elimination

The elimination method is used to solve systems of equations by eliminating a variable and determining the value of the variable to find the solution. Given below is an image showing the application of the elimination method to solve a system of equations with two variables. Consider two equations x - 2y = 8 and 2x + y = 5. 

Steps to Use Elimination Method

The elimination method is useful to solve linear equations containing two or three variables. We can solve three equations as well using this method. But it can only be applied to two equations at a time. Let us look at the steps to solve a system of equations using the elimination method.

• Step-1: The first step is to multiply or divide both the linear equations with a non-zero number to get a common coefficient of any one of the variables in both equations.
• Step-2: Add or subtract both the equations such that the same terms will get eliminated.
• Step-3: Simplify the result to get a final answer of the left out variable (let's say, y) such that we will only get an answer in the form of y=c, where c is any constant.
• Step-4: At last, substitute this value in any of the given equations to find the value of the other given variable.

These are the elimination method steps to solve simultaneous linear equations. Let us take an example of two linear equations x+y=8 and 2x-3y=4 to understand it better.

Let, x + y = 8 → (1) and 2x - 3y = 4 → (2)

Step 1: To make the coefficients of x equal, multiply equation (1) by 2 and equation (2) by 1. We get,

(x+y=8) × 2 → (1)

(2x-3y=4) × 1 → (2)

So, the two equations we have now are 2x + 2y = 16 → (1) and 2x - 3y = 4 → (2).

Step 2: Subtract equation 2 from 1, we get, y=12/5.

Step 3: Substitute the value of y in equation 1, we get, x + 12/5 = 8

x = 8 - 12/5

x = 28/5

Therefore, x = 28/5 and y=12/5.

But what if while multiplying a non-zero constant, we get the coefficients of both the variables equal? What if both the terms got eliminated while adding or subtracting? We get such cases while solving equations of parallel and coincident lines. Equations of two intersecting lines will have only two solutions that are consistent, but the equations of two parallel lines have no solutions as these lines never intersect each other. And the equations of coincident lines have infinitely many solutions as they lie on each other so every point is the intersection or the common point of those lines. Let us discuss each of these two cases in detail.

Elimination Method: No Solutions

As we know that equations of two parallel lines have no solutions. So, if we solve any such equations using the elimination method we get the answer as two unequal numbers on both sides of the unequal sign. For example, 0≠8, -2≠0, etc. In such cases, we cannot eliminate only one variable. Both the variables get eliminated. For example, let us solve two equations 2x - y = 4 → (1) and 4x - 2y = 7 → (2) by the elimination method. In order to make the x coefficients equal in both the equations, we multiply equation (1) by 2 and equation (2) by 1. By doing so we get, 4x - 2y = 8 → (3) and 4x - 2y = 7 → (4). Now, if we try to subtract equation 4 from equation 3, we get, 0=1 as both the variables are getting eliminated. There is no other way to solve these equations as the solutions are inconsistent. So, in the elimination method when there is no solution, we get the result in this form.

Elimination Method: Infinitely Many Solutions

Two equations of coincident lines have infinitely many solutions possible. So, if by the elimination method we solve a system of equations of coincident lines, we get a consistent system with infinite solutions. In such cases, we get an answer in the form of 0=0 if we apply the elimination method. For example, try to solve equations x+y=2 and 2x+2y-4=0. If you multiply any non-zero constant with both equations, you will find that every time x-variable terms and y-variable terms are getting canceled or eliminated. So, in the elimination method when there are infinitely many solutions possible, we get the result in the form of 0=0. It is advisable to check whether the given linear equations are of intersecting lines, parallel lines, or coincident lines before trying to solve them. Check this article to know about the solutions of linear equations.

Solving System of 3 Equations Using Elimination Method

To solve a system of three linear equations with the elimination method, first, make sure that the equations are written in the standard form Ax+By+C=0 or Ax+By=C without any fractional coefficient. Take any two equations as per your comfortability and select a variable to eliminate. Eliminate the chosen variable. Now, select another pair of equations out of the given three equations and eliminate the same variable. This way you will get two equations with only two variables. Solve those using the elimination method steps mentioned above and find the values of those 2 variables. Substitute the values in any of the given equations to find the value of the third variable.

Let's solve three equations 3x-y+2z=5, 4x+2y-z=6, and 5x-3y+z=1 for a better understanding.

Now, we have found that x=1. Substitute this value in equation P, we get 9×(1)-y=7.

9-y=7

y=2

Now put the values of x and y in the third equation 5x-3y+z=1, and we get z=2. Therefore, x=1, y=2, and z=2.

Important Notes on Elimination Method

• The elimination method is used to solve a system of equations.
• This method is easy and makes the calculations easier as it eliminates one variable and hence, reduces the calculations.
• We make the coefficient of a variable identical to eliminate the corresponding variable.

Related Articles

• Elimination Method Calculator
• Substitution Method
• System of Equations Solver
• Cross Multiplication Method
• Graphically Solving a Pair of Linear Equations
• Solving Linear Equations

FAQs on Elimination Method

What is the Elimination Method in Math?

In math, the elimination method is used to solve a system of linear equations. It is the most widely used and simple method as it involves fewer calculations and steps. In this method, we eliminate one of the two variables and try to solve equations with one variable. The value found here can be substituted in any of the given equations to find the value of the other variable as well.

How do you Solve Linear Equations Using the Elimination Method?

Simultaneous linear equations can be solved using the elimination method. First of all, make sure that the equations are written in the standard form either Ax+By=C or Ax+By+C=0. In this method, we multiply both the equations with a non-zero number to make the coefficients of any one variable equal. Then we add or subtract the equations to eliminate one of the variables to find the value of the other variable. This is how we solve linear equations by the elimination method.

What are the Steps Involved in the Elimination Method?

The steps involved in the elimination method are given below:

• Choose any one variable to eliminate. Multiply or divide both the equations with a non-zero constant to make the coefficients of that variable equal.
• Add or subtract the resultant equations such that the chosen variable gets eliminated.
• Simplify and find the value of the other variable.
• Substitute that value in any of the given equations to find the value of the eliminated variable.

Can the Elimination Method be Used to Solve the System of Equations in Three Variables?

Yes, the elimination method can be used to solve linear equations with three variables. With three equations, we take any two equations and select the variable to be eliminated. Then we repeat the same process by taking another pair of equations and eliminate the same variable. In this way, we will be left with two equations in two variables only that can be solved using the elimination method. At last, we put the values of those two variables in any of the given equations to find the value of the third variable.

When Should you Use the Elimination Method?

It is better to use the elimination method when the coefficients of any one variable in the equations are the same. For example, 3x+7y+2=0 and 3x-4y+5=0. The other methods of solving linear equations are substitution method, cross multiplication method, graphical method, and matrix method.

What is the Difference Between the Elimination Method and the Substitution Method?

The difference between the substitution and the elimination method is that in the substitution method we find the value of one variable in terms of the other variable. Then, we substitute that value in the second equation to find the value of the other variable. But in the elimination method, we eliminate any one variable and then find the value of the other variable.

How Do You Solve Systems of Equations Using the Elimination Method?

To solve systems of equations using the elimination method, we make the coefficient of one of the variables involved identical and then add/subtract the equations in order to eliminate that variable and find the value of the other variable.

Why is Elimination Method Better?

The elimination method is considered better than other methods for solving a system of equations because it makes the calculations easier by eliminating a variable. Once a variable is eliminated, it becomes easier to determine the value of another variable and then use this value to find the value of the eliminated variable.