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# What is the Elimination Method? Explained with the help of examples

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## What is the Elimination Method? Explained with the help of examples

In mathematics, the elimination method is a technique for solving a system of linear equations. Algebraically, elimination method is the most commonly used method out of all the methods to solve linear equations.

In this article, we will learn how to calculate systems of equations using the elimination method. An elimination method is chosen over the substitution method when it is simply to multiply the coefficient and add or subtract the equations to eliminate one of the variables.

## Definition of Elimination Method:

An elimination method is a technique used to solve a system of linear equations. In this method, we manipulate the equations in the system by adding, subtracting, or multiplying them by constants so that one of the variables is eliminated.

In the elimination method, we eliminate any one of the variables using basic arithmetic operations and then calculate the equation to solve the value of the other variable.

Now, we can put that value in any of the equations to find the value of the variable eliminated. The system of linear equations is solved as a set of two or more linear equations with two or more unknown variables. Then, solving the linear equation is calculating the solutions of the unknown variables in the method of equations.

A different procedure to calculate the solutions for the unknown variables. They are the graphical method, substitution method, elimination method, cross-multiplication method, and so on. Generally, the elimination method is easy to use because here we eliminate one of the terms that make our simple calculations.

In mathematics, a system of the linear equation can be written as,

Ax + By = C → (1)

Px + Qy = R → (2)

## Solving system of linear equation through Elimination method

An elimination method is one of the procedures to calculate the system of linear equations. Now, in this method, either add or subtract the equations to get the equation in one variable. Generally, if the coefficients of one of the variables are the same, and the sign of the coefficients is opposite, we can add the equation to eliminate the variable.

Similarly, if the coefficients of one of the variables are the same, and the sign of the coefficients is the same, we can subtract the equation to get the equation in one variable. In this case, if we do not have the equation to directly add or subtract the equations to eliminate the variable.

Now, you can begin by multiplying one or both equations by a constant value on both sides of an equation to obtain the equivalent linear system of equations. Then they eliminate the variable by simply adding or subtracting the equations.

## Steps to solve the Elimination Methods:

The elimination method to calculate a system of linear equations containing two or three variables. In this case, we can calculate three equations as well using the elimination method. Generally, you can only be applied to two equations at a time. Let’s suppose look at the steps to solve a system of linear equations using the elimination method.

Step 1:

The first step is to multiply or divide both 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 equations such that the same terms will get eliminated.

Step 3:

The result is simply to get a final solution of the left-out variable, such that we will only get a solution 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 calculate the value of the other given variable.

## How to solve system of linear equations?

In this section, we explain the elimination method with the help of examples.

Example 1:

Consider an example of two linear equations x+y=8 and 2x-3y=4.

Let,

x + y = 8 →(1)

2x – 3y = 4 → (2)

Solution:

Step 1:

To make a coefficient of x equal, multiply equation (1) by 2 and equation (2) by 1. We get

(x+y =8) X 2 →(1)

(2x-3y =4) X 1 →(2)

So, the two equations we have now are,

2x+2y=16 →(1)

2x-3y=4 →(2)

Step 2:

Now, we subtract equation (2) from (1), and we get y = 12/5.

2x+2y =16

2x-3y =4

After subtracting it becomes,

5y = 12

Y = 12/5

Step 3:

Now, we subtract the value of y in equation (1), and we get, x+12/5=8.

x= 8-12/5

x= 28/5

Therefore, x= 28/5 and y= 12/5 or (28/5, 12/5).

You can also take assistance from an elimination method calculator to find the values of unknown variables of system of linear equations with steps.

Example 2:

Solve the system of linear equations,

2x+3y=8 →(1)

3x+2y=7 →(2)

Solution :

Step 1:

Multiply each equation by a suitable number so that two equations have the same lading coefficient. An easy choice in multiply equation (1) by 3, then the coefficient of x in equation (2), and multiply equation (2) by 2 then the x coefficient in equation (1).

(2x+3y= 8) X 3 →(1)

6x+9y= 24

(3x+2y= 7) X 2 →(2)

6x+4y= 14

Step 2:

Now, we subtract the second equation from the first equation.

6x+9y= 24

6x+4y= 14

After subtracting it become,

5y = 10

Step 3:

Solve this new equation for y.

Y = 10/5

Y = 2

Step 4:

Substitute y = 2, into either equation (1) or equation (2), above and solve x. We will use equation (1).

2x+3(2)= 8

2x+6= 8

2x= 2

x= 1

Therefore, x=1 and y=2 or (1,2).

## Conclusion

In this article, we’ve discussed the basic definition of the Elimination Method, solving the system of linear equations by using the Elimination Method, and steps to solve the Elimination Method. Additionally, with the help of examples explained the topic in detail.

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