# Elimination Method: How to Solve System of Linear Equation

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The elimination method is a mathematical technique used to solve systems of linear equations. It involves manipulating the equations in such a way that one variable is eliminated, leaving a simpler equation that can be solved for the remaining variable.

The method has a long history, dating back to ancient civilizations, and it is still widely used today in a variety of fields, including engineering, physics, and finance. Further in this article, we will study the definition, basic history, and some real-life life uses of the elimination method.

## What is the elimination method?

The elimination method is a logical method to identify an entity of interest among several ones by excluding all other entities. It is a well-known way to solve the system of linear equations. In this method, the linear equations will be solved simultaneously to find the unknown values of the equation.

### History of the Elimination Method:

The earliest known use of the elimination method can be traced back to ancient Babylon, where it was used to solve systems of linear equations in one variable. The method was also used by the ancient Egyptians and Greeks, and it was later adopted by the Chinese and the Islamic world.

In the 16th century, the German mathematician and astronomer Johannes Kepler used the method to solve systems of linear equations in two variables. He used it to calculate the positions of the planets in the solar system, which helped to improve the accuracy of astronomical predictions.

During the 18th and 19th centuries, the elimination method was further developed by several mathematicians, including Gauss, Cramer, and Lagrange. They introduced new techniques for solving systems of linear equations, such as Gaussian elimination and Gauss-Jordan elimination.

In the 20th century, the elimination method was further refined and extended to systems of linear equations with more than two variables. The advent of the electronic computer in the 1940s and 50s made it possible to solve large systems of equations quickly and efficiently, which increased the method’s popularity in fields such as engineering, physics, and finance.

### Real-Life Applications of the Elimination Method:

The elimination method is used in a wide variety of fields, including engineering, physics, and finance. Some examples of its real-life applications include:

#### Engineering:

Engineers use the elimination method to solve systems of linear equations that describe the behavior of mechanical systems, electrical circuits, and other types of engineering systems. For example, an engineer may use the method to calculate the forces acting on a bridge or the voltage and current in an electrical circuit.

#### Physics:

Physicists use the elimination method to solve systems of linear equations that describe the behavior of physical systems, such as the motion of particles and the properties of materials. For example, a physicist may use the method to calculate the trajectories of particles in a particle accelerator or the properties of a crystal.

#### Finance:

Financial analysts use the elimination method to solve systems of linear equations that describe the behavior of financial systems, such as the prices of stocks and bonds. For example, an analyst may use the method to calculate the fair value of an option or the yield on a bond.

In the next section, we are going to elaborate on the elimination method with the help of mathematical examples.

### Example section:

Example 1:

Solve the system of linear equations 3x + 2y = 6; x – 3y = 2 using the elimination method.

Solution:

3x + 2y = 6 −−−−−−−− (1)

x − 3y = 2 −−−−−−−− (2)

Step 1: Simplify Eq (1) and Eq (2)

3x + 2y − (6) = 0

x − 3y − (2) = 0

3x + 2y – 6 = 0

x − 3y – 2 = 0

3x + 2y = 6

x − 3y = 2

Step 2: Find coefficients of x.

coefficients of x are [3, 1]

So, multiply Eq (1) by 1 and Eq (2) by 3

3x + 2y = 6

3x − 9y = 6

Step 3: Subtraction.

Now subtract Eq (1) and Eq (2)

3x + 2y = 6

−3x + 9y = 6​

11y = 0

y = 0 / 11 ​

y = 0

Step 4: Insert the value of “x”.

Now replace the value of y in 3x + 2y = 6

3x + 2(0) = 6

3x = 6

x = 6 / 3

x = 2

The solution set is {2, 0}

You can take assistance from the elimination calculator by Allmath (https://www.allmath.com/elimination-calculator.php) to solve the system of linear equations without any difficulty.

Example 2:

Solve the system of linear equations 2x + 3 = -5y; 6y – 2 = 23x using the elimination method.

Solution:

2x + 3 = −5y −−−−−−−−(i)

6y – 2 = 23x −−−−−−−−(ii)

Step 1: Simplify Eq (1) and Eq (2)

2x + 3 − (−5y) = 0

6y – 2 − (23x) = 0

2x + 5y + 3 = 0

−23x + 6y – 2 = 0

2x + 5y = −3

−23x + 6y = 2

Step 2: Find coefficients of x.

coefficients of x are [2, −23]

So, multiply Eq (1) by -23 and Eq (2) by 1

−46x − 115y = 69

−46x + 12y = 4

Step 3: Subtraction.

Now subtract Eq (1) and Eq (2)

− 46x − 115y = 69

−46x + 12y = 4

We get:

−127y = 65

y = −65 / 127

y = −0.5118

Step 4: Insert the value of “x”.

Now replace the value of y in 2x + 5y = −3

2x + 5(−0.5118) = −3

2x − 2.5590 = −3

2x = −3 + (2.5590)

x = −0.4409 / 2​

x = −0.2205

The solution set is {−0.2205, −0.5118}

### Conclusion:

In this article, we have studied the elimination method which is a powerful technique for solving systems of linear equations, we have covered topics like the history of the elimination method, and we have also studied its real-life uses and applications in real-life.

In the example section we have elaborated on the step-by-step solution, you have witnessed that it is not a difficult topic now you can easily solve the system of linear equations using the elimination method.

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