# 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

**Step 1:**

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

**Step 2:**

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

**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

When two numbers are subtracted,
the result is four.

or

**Step 3:**

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.