# Gaussian Elimination Calculator with steps

## Gaussian Elimination Calculator

Our Gaussian Elimination Calculator with steps is a powerful tool for solving systems of linear equations. It can handle up to 10 variables and provides a step-by-step solution that helps you understand the calculations involved.

To use the Gaussian Elimination Calculator, follow these steps:

1. Resize the matrix according to the number of equations in your system, using the + – buttons.
2. Enter the coefficients in the corresponding fields. You can also change the variable names, as they are named x1, x2,…,xn by default.
3. Press the Calculate button to get a step-by-step solution.
4. Follow along with each calculation and see how the solution is derived.

Our calculator is a great resource for students, teachers, and professionals alike. Give it a try and simplify your linear algebra computations!

## What is Gaussian elimination method?

Gaussian elimination is a method used to solve systems of linear equations by transforming the system into an equivalent upper triangular matrix. This process involves performing elementary row operations such as adding a multiple of one row to another or interchanging two rows. The goal is to simplify the system of equations so that it can be easily solved using back-substitution. Gaussian elimination is a widely used algorithm in mathematics, particularly in linear algebra. It is taught in many undergraduate mathematics courses and is an important tool for solving systems of linear equations with many variables. One of the advantages of Gaussian elimination is that it can be easily implemented in computer algorithms, making it useful for solving large-scale systems of equations in a variety of fields. Overall, Gaussian elimination is a powerful method for solving systems of linear equations that has a wide range of applications in mathematics and other fields.

## How to do Gaussian elimination

To perform Gaussian elimination on a system of linear equations, follow these steps:

1. Write the augmented matrix of the system, which consists of the coefficients and the constants of the equations.
2. Choose a pivot element, which is the first non-zero element in the first row.
3. Use elementary row operations to make all the elements below the pivot in the same column equal to zero. This is done by subtracting multiples of the first row from the subsequent rows.
4. Move to the next row and repeat steps 2 and 3 until the matrix is in upper triangular form.
5. Solve the system by back-substitution. Start with the last row and solve for the variable corresponding to the pivot element. Substitute that value into the second-to-last row and solve for the corresponding variable, and so on until all variables have been solved for.

It is important to note that if a pivot element is zero, you must interchange that row with another row below it that has a non-zero element in the same column before proceeding with the elimination process.

Following these steps will allow you to solve systems of linear equations using Gaussian elimination.

## Example of Gaussian elimination

 5x+2y = 17 4x+5y = 8

Solution by Gaussian elimination

Convert the augmented matrix into the row echelon form:
$\underset{}{\left(\begin{array}{ccc}\overline{)5}& 2& 17\\ 4& 5& 8\end{array}\right)}$
$×\left(-0.8\right)$
$\stackrel{}{\underset{{R}_{2}-\left(0.8\right)\cdot {R}_{1}\to {R}_{2}}{~}}$$\underset{}{\left(\begin{array}{ccc}5& 2& 17\\ 0& 3.4& -5.6\end{array}\right)}$
• Find the variable $y$ from the equation $2$ of the system :
$3.4\cdot y=-5.6$
$y=-1.647$
• Find the variable $x$ from the equation $1$ of the system :
$5x=17-2y=17-2\cdot \left(-1.647\right)=20.294$
$x=4.059$
• $x=4.059$
• $y=-1.647$
General Solution: $X=\underset{}{\left(\begin{array}{c}4.059\\ -1.647\end{array}\right)}$

(This example has been generated with the Gaussian Elimination Calculator)

## Applications

Here are some of the applications of the Gaussian elimination method:

1. Solving systems of linear equations: The Gaussian elimination method is commonly used to solve systems of linear equations. This is particularly useful in engineering and science applications, where linear equations are used to model physical systems.

2. Inverse matrix: The inverse matrix of a matrix can be computed using Gaussian elimination. This is useful in many applications, such as cryptography and optimization problems.

3. Eigenvalues and eigenvectors: The Gaussian elimination method can be used to compute the eigenvalues and eigenvectors of a matrix. This is important in many fields, such as physics and engineering, where eigenvalues and eigenvectors are used to study the behavior of physical systems.

4. Linear programming: Gaussian elimination can be used to solve linear programming problems. This involves maximizing or minimizing a linear objective function subject to linear constraints.

5. Signal processing: Gaussian elimination is used in signal processing applications to solve linear systems of equations that arise in the analysis of signals.

6. Computer graphics: Gaussian elimination is used in computer graphics to solve linear systems of equations that arise in the rendering of 3D graphics.

7. Circuit analysis: Gaussian elimination is used in circuit analysis to solve systems of linear equations that arise in the analysis of electrical circuits.