# What is Elimination Method – Steps, Techniques, and Examples

The elimination system is a procedure used in eliminating one variable by manipulating the equations in such a way that when they are added or subtracted, one of the variables is eliminated, leaving only one variable to solve for.

The elimination method works by taking two equations with two variables and rearranging them so that one of the variables is eliminated when the two equations are added or subtracted. To do this, multiply one equation by the required number to make both numbers equal.

In this article, we will discuss the definition of the Elimination Method, the Method of elimination History of Steps used to solve the Elimination Method, and also explain the topic with examples.

## What is the elimination method?

The elimination method, also known as the addition method or the linear combination method, for solving linear equations we use this entails eliminating one of the factors in a system by subtracting or adding equations, then solving for the remaining variable.

The goal is to reduce the system of equations to a simpler form with fewer variables and eventually arrive at a solution for each of the variables in the system. The elimination method is particularly useful when solving systems of equations with two variables, but can be extended to larger systems as well.

### How to Solve a System of Equations Using the Elimination Method?

To solve a system of equations, rewrite the equations so that one or two variables can be removed when these two equations are added or subtracted. The aim is to rewrite the equation so that it will be easier for us to eliminate the terms.

Step 1: First write the main equation in its original form

Both equations should be written in standard form, with the variables on one side of the equation and the constants on the other

Step 2: Select a variable to find

Select the variable that will result in the most straightforward computations.

Step 3: Multiply the needed equations

To ensure that one of the factors is removed when the equations are added or subtracted, multiply one or both equations by a constant.

Step 4: Add or subtract both equations

To take the enquired variable out, subtract from or add to the two calculations.

Step 5: Enter the discovering variable's value once more.

To discover the value of the other variable, once more enter the value of the finding variable into one of the original equations. In this instance, we can change the first equation's value of x to equal 2:

Step 6: Check the solution

Test the solution after putting in the x and y values. The solution is valid if the results of the two formulae are accurate.

Elimination process History

The elimination method was also independently discovered in Europe during the 17th century. The French mathematician Rene Descartes is credited with the first known use of the method in Europe, where he used it to solve a system of three equations with three unknowns.

During the 18th and 19th centuries, the elimination method became a widely used technique in algebra and was taught in schools and universities. The method was used to solve a variety of problems, including those in physics, engineering, and economics.

With the development of computers and numerical methods in the 20th century, the elimination method has been largely replaced by other techniques, such as Gaussian elimination and LU decomposition. However, the elimination method remains an important tool in algebraic theory and is still taught in many mathematics courses today.

## Examples of Elimination Method

Ex# 1:

Solve it by using the elimination method

6p−2q=32

3p+q=8

Find the value of p and q using the elimination method

Solution:

Given equations are

6p – 2q = 32               ←      (1)

3p + q = 8                   ←      (2)

Step 1:

For this, multiply equation (2) by 2 to easily make the coefficients of the single variable term equivalent. We'll obtain,

6p−2q=32                   ←       (1)

6p+2q=16                   ←       (2)

Step 2:

To locate the q variable term, add the two formulae.

Divide 12 into both sides

p = 4

Step 3:

Now put the value of p in the equation in 1

Equation 1 is

6p−2q=32

6(4)-2q=32

24-2q=32

-2q=32-24

-2q=8

Dividing -2 into both sides

q = -4

Therefore, p = 4 and q = -4

Example 2

The outcome of adding two numbers and reversing one of them is 110. Decide how many numbers differ by four. What is the entire number?

Solution:

According to the given condition, we solve the question step by step

Step 1:

Let the 10 and the unit’s digits in the first number be y and x, respectively.

So, the first number = 8x + y

And then the reversed digits

The second number will be = x + 8y

Step 2:

As per the given statement;

(8x + y) +(8y + x) = 110

9x + 9y = 110

9(x + y) = 110

x + y = 10                ←          (1)

When two numbers are subtracted, the result is four.

x – y = 4                   ←          (2)

or

y – x = 4                   ←          (3)

Step 3:

Adding equations (1) and (2) we get

Find the value of x=7

x=7

put value x in the equation in (1)

x + y = 10

7+y=10

y=3

x = 7 and y = 3

Hence, the number is 59.

Step 4:

If we consider equations 1 and 3, then by elimination method we get,

x=3 and y=7

Hence, the number is 37.

Therefore, there are two such numbers, 59 and 37.

The unknown terms of the system of linear equations can also be determined with the help of an elimination method calculator to avoid time-consuming calculations.

## Summary

In this article, we have discussed the definition of the Elimination Method, the Steps used to solve the Elimination Method, and the Method of elimination History also explain the topic with example. and also with the help of an example topic will be explained. After studying this article everyone can defend this topic.

Courtesy : CBSE