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

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.