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Math Calculators ▶ Gaussian Elimination Calculator

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**Table of Content**

This free Gaussian elimination calculator is specifically designed to help you in resolving systems of equations. Yes, now getting the most accurate solution of equations is just a couple of clicks away.

In the light of mathematical analysis:

**“The particular method that is used to find solution to the linear equations by arranging the augmented matrix of their coefficient numbers is known as the Gaussian Algorithm”**

Here we are going to apply this theorem on an example below. So for better understanding, just stay focused!

**Example # 01:**

Find solution of the following system of equations as under:

$$ 3x_{1} + 6x_{2} = 23 $$

$$ 6x_{1} + 2x_{2} = 34 $$

**Solution:**

No doubt our widely used gaussian elimination calculator with steps will show detailed calculations to simplify these equations, but we need to analyse the scenario manually.

The equivalent augmented matrix form of the above equations are as follows:

$$ \begin{bmatrix} 3&6&23 \\ 6&2&34 \\\end{bmatrix} $$

Gaussian Elimination Steps:

**Step # 01:**

Divide the zeroth row by 3.

$$ \left[\begin{array}{cc|c}1&2& \frac{23}{3} \\6&2&34 \\\end{array}\right] $$

**Step # 02:**

Multiply the first row by 6 and then subtract it from the zeroth row.

$$ \left[\begin{array}{cc|c}1&2&\frac{23}{3} \\0&-10&-12 \\\end{array}\right] $$

**Step # 03:**

Go for dividing the first row by -10.

$$ \left[\begin{array}{cc|c}1&2&\frac{23}{3} \\0&1&\frac{6}{5}\\\end{array}\right] $$

**Step # 04:**

Get going for finding the product of zeroth row and 2. After doing that, subtract the result from the first row.

$$ \left[\begin{array}{cc|c}1&0&\frac{26333333334}{5000000000}\\0&1& \frac{6}{5}\\\end{array}\right] $$

As you see on the left side of the matrix, we get the identity matrix. So the answer on the right side pof the equation would be the values of the variables in the equations.

So the final results are as follows:

$$ b_{1} = 5.266 $$

$$ b_{2} = 1.2 $$

The same results can also be verified by using outer free gauss elimination calculator.

Get going to understand how this free gaussian elimination solver matrix row reduction algorithm simplifies equation systems.

**Input:**

- First, set up the order of the matrix from drop-down lists
- After you do that, click the “Set Matrices” button to get the desired matrix format
- Now fetch the numbers in their fields
- After you are done with the stuff, hit the calculate button

**Output:**

The best gauss jordan elimination calculator with steps does the following calculations:

- Shows variables’ coefficients
- Displays Gaussian elimination steps

From the source of Wikipedia: Gaussian elimination, Row operations, Echelon form, Computing determinants, inverse of a matrix, Ranks