Introduction to the elimination method with its real-life
application.

The elimination method is used for solving a system of linear equations from an algebraic point of view it is mostly used as compared to all other methods.

In this
method, one variable is eliminated to get the answer of another variable after
simplifying. Then to get the answer of another variable put the required
variable answer to any of the equations.

It is an
easy method because in this when a variable is eliminated it makes calculation
easy. After it, you can easily get results of an unknown variable because it
converts the equation into a single variable then after applying the basic
technique of equation solving you can evaluate your unknown variables.

Furthermore,
in this article, the basic idea of the elimination method and methods to solve
systems of equations using this elimination method technique with help of
different examples based on this concept will discuss.

## What is the elimination method?

It is an
analytical process to choose an unknown variable among all by excluding all
other entities.

It is
basically a method to solve a system of linear equations by multiplying or
dividing a number by the equations to make the coefficients the same as any one
variable.

Then after
making the coefficient of the variable same then for its cancelation check addition
or subtraction is required. For this reason, the
addition method is another name for the elimination method.

## Use of the Elimination method for
real-life simultaneous equations

There are
a few scenarios where we can use the elimination method to solve everyday life
simultaneous equations.

·
**Rate, Distance, and Time**

You can
find the beneficial route for running or other sports by using a mathematical
expression that considers average speed and distance for a different part of
the route also you can use other equations to set your routine and get benefit
from it by maximizing time or your speed for good performance.

·
**Planes, Trains, and Automobiles**

If you
want to know the values for the unknown variables in your traveling situation
like to determine distance, speed, or time taken according to your source like
by car, plane, or train used during traveling and also you can calculate
running time.

·
**The Best Deal**

You can
check the Best Deal if you want to find out a car for rent, and you're
comparing two rental companies.

## Example Section:

In this
section with the help of examples, the topic will be explained for better
understanding.

**Example 1:**

Evaluate
these simultaneous linear equations:

**3x + 2y = 3; 4x
+ y = 2**

**Solution:**

**3x + 2y =
3 Equation 1**

**4x + y = 2 Equation 2**

**Step
1: **Making like coefficients

What we do
in 1^{st} step is to make the coefficients same of any single variable
of both equations here if we multiply 4 by equation 1 and 3 by equation 2 in result,
we get the same coefficient
of variable x.

**For
equation 1:**

4* (3x + 2y
= 3)

12x + 8y =
12

**For
equation 2: **

3 * (4x +
y = 2)

12x + 3y =
6

**Step
2: **Subtraction.

Subtract
the second equation from the first.

12x + 8y =
12

-(12x + 3y
= 6)

5y
= 6

**Step
3: **Simplify

**y = 6/5 **

**Step
4: **Calculate the value of “x”

Put the
value of y i.e., 6/5 into any equation either in equation 1 or equation 2.

3x +
2(6/5) = 3

15x + 12 =
15 Multiply by 5 on both sides

15x = 15
-12

15x = 3

x = 3/15

**x = 1/5**

**{x, y} =
{6/15, 1/5}**

**Example**
**2:**

Evaluate
these simultaneous linear equations:

**4x + 5y = 4**;
**2x + y = 1**

**Solution:**

**4x + 5y =
4 Equation 1**

**2x + y = 1 Equation 2**

**Step
1: **Making like coefficients

**For equation 1:**

2 * (4x +
5y = 4)

8x + 10y =
8

**For
equation 2:**

4 * (2x +
y = 1)

8x + 4y = 4

**Step
2: **Subtraction.

By subtracting
equation 2 from equation 1 we get:

8x + 10y = 8

-(8x + 4y
= 4)

6y = 4

**Step
3: **Simplify

y = 4/6

**y = 2/3 **

**Step
4: **Calculate the value of “x”

Putting y
= 2 / 3 in equation 1.

4x +
5(2/3) = 4

12x + 10 =
12

12x = 2

x = 2/12

**x = 1/6 **

**{x, y} = (1/6,
2/3)**

Sometimes
we get bored of solving the equations simultaneously, to get rid of these
lengthy calculations you can take assistance from the Elimination
calculator by Allmath and you can also save your valuable
time.

**Example**
**3:**

Solve
these equations and find out the values of x and y.

**-9x - 2y =
11**; **13x + 7y = -31**

**Solution:**

**-9x - 2y =
11 Equation 1**

**13x + 7y =
-31 Equation 2**

**Step
1: **Making like coefficients

**For
equation 1:**

Before
making the coefficients same re-arrange the equation.

-9x - 2y =
11

0 = 11 +
9x + 2y

9x + 2y =
- 11

13 * (9x +
2y = -11)

117x + 26y
= -143

**For
equation 2:**

9 * (13x +
7y = -31)

117x + 63y
= -279

**Step
2: **Subtraction.

Subtract
equation 2 from equation 1

117x + 26y = -143

-(117x +
63y = -279)

-37y = 136

**Step
3: **Simplify

y = -136/37

**Step
4: **Calculate the value of “x”

Putting y
= -136/39 in equation 1.

-9x –
2(-136/37) = 11

-9x +
2(136/37) = 11

-9x +
272/37 = 11

-333x +
272 = 407

-333x =
407 – 272

-333x =
135

**x = -135/333
**

**{x, y} = {-136/37,
-135/333}**

## Summary:

In this
article, you have studied the basic role of the elimination method and its use in
solving the system of linear
equations.

Moreover,
with the help of examples ways to solve the system of linear equations are
discussed. After thoroughly reading and understanding this article you can
easily defend the questions related to this topic.

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