The elimination method is commonly used to solve linear algebraic equations. In this method, we eliminate one variable from equations by using basic arithmetic operations and then simplify equations to find the values of other variables.

In this article, we study what is elimination method, what are the importance of the elimination method and also discuss some examples for better understanding.

**What is the elimination method in algebra?**

The method of solving linear equations by eliminating one variable and simplifying equations to find out the values of other variables is called the __elimination method__.

**What is the System of the linear equation:**

A system of linear equations, when two or three equations with two or three variables are given is called a system of linear equations.

**Examples:**

**Example 1:**

2x + 3y = 9

4x – 3y = 8

**Example 2:**

x + y + z = 0

5x + 3y – z = 7

2x – y + 7z = 0

**Steps of Elimination method:**

There are the following steps for __solving a system of equations by elimination method__:

**Step 1:**To get the common coefficient of one variable of both equations we multiply or divide by a non-zero number.**Step 2:**We add or subtract both equations to eliminate the same variable with the same coefficient.**Step 3:**Then simplify the remaining equation to get out the final result of another variable.**Step 4:**Finally, we substitute the given value in any other equation to find out the value of the other variable given.

**Solving System of linear equations:**

There are the following types of solving systems of linear equations:

- System of linear equations with No solution
- System of linear equations with three equations and three variables
- A system of linear equations with two equations and two variables is given
- System of linear equations with Infinity many solutions

### 1. **A system of linear equations with two equations and two variables is given**

There is an example of a System of linear equations with two equations and two variables given in the Example section.

### 2. **System of linear equations with three equations and three variables**

A system of linear equations with three equations and three variables also solve by the same method.

### 3. **System of linear equations with No solution**

We know that parallel lines have no solution, if we solve such problems can assume that all variables eliminate and we get some unequal numbers on both sides of the unequal sign. Like as -2 ≠ 0 or 5 ≠ 7

**Example:**

X + Y = 0

X + Y = 7

### 4. **System of linear equations with Infinitely many solutions**

We know that coincident equations of lines have infinitely many solutions. If we try to solve the system of coincident lines we get a consistent system with infinite solutions. In this case, we get an answer in the form of

0 = 0.

**Example:**

x + y = 4

2x + 2y – 8 = 0

**Examples of Elimination Method**

**Example 1:**

2x + 3y = 9

4x – 2y = 8

**Solution:**

2x + 3y = 9 ………. (i)

4x – 2y = 8 ………. (ii)

**Step 1:** Multiply or divide by a non-zero number to get a common coefficient of one variable

4 multiply by Eq. (i)

4 x (2x + 3y = 9)

8x + 12y = 36 ………. (iii)

2 multiply by Eq. (ii)

2 x (4x – 2y = 8)

8x – 4y = 16 ………. (iv)

**Step 2:** Add or subtract both equations to eliminate the same variable with the same coefficient

Subtract Eq (iv) from Eq (iii)

(8x + 12y) – (8x – 4y) = (36) – (16)

**Step 3:** Simplify the remaining equation to get out the final result

8x – 8x + 12y + 4y = 36 – 16

16y = 20

y = 20 / 16

**y = 5 / 4 **

**Step 4:** Substitute the given value in any other equation to find out the value of the other variable given

Put y = 5 / 4 in Eq. (i)

2x + 3 * (5 / 4) = 9

2x + (15 / 4) = 9

2x = 9 – (15 / 4)

2x = (36 – 15) / 4

2x = 21 / 4

x = 21 / (4 * 2)

**x = 21 / 8 = 2.625**

Alternatively, you can get help from online calculators to evaluate the problems easily in no time. Let us take the above example to __solve by elimination calculator__. It will give us the result in a couple of seconds.

**Example 2:**

x – 2y = 8

2x + y = 5

**Solution:**

x – 2y = 8 …………(i)

2x + y = 5 …………(ii)

**Step 1:** Multiply or divide by a non-zero number to get a common coefficient of one variable

2 multiply by Eq. (ii)

2 x (2x + y = 5)

4x + 2y = 10 …………(iii)

**Step 2:** Add or subtract both equations to eliminate the same variable with the same coefficient

Add Eq. (i) and Eq. (iii)

(x – 2y) + (4x + 2y) = (8) + (10)

**Step 3:** Simplify the remaining equation to get out the final result

x + 4x – 2y + 2y = 8 + 10

5x = 18

**x = 18 / 5**

**Step 4:** Substitute the given value in any other equation to find out the value of the other variable given

Put x = 18 / 5 in Eq. (i)

(18 / 5) – 2y = 8

-2y = 8 – (18 / 5)

-2y = (40 – 18) / 5

-2y = 22 / 5

y = 22 / (5 * -2)

y = – 22 / 10

**y = – 11 / 5**

**Importance of elimination method:**

There is the following important elimination method:

- The elimination method is used to solve a system of the linear equation of two or three equations with two or three variables.
- It is an easy method to solve linear equations.
- It reduces the calculation to solve a linear equation.
- In this method, we make the variable coefficient identical to eliminate the corresponding variable.

**Summary:**

In this article, we discuss the elimination method and its type of solving system of linear equations with two variables and with three variables of two equations and three equations. We discuss what are the system of linear equations and the system of linear equations with no solution and with infinitely many solutions.