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Gaussian Elimination - Solving any systems of equation - calculator

You can calculate with explanations any system of linear equations, both homogeneous and heterogeneous with any number of unknowns by Gaussian elimination method. As a result, apart from the solution, you will also receive a complete analysis and a step-by-step calculation.
To get the result, enter the coefficients in empty cells or by clicking on , change to the text field and enter the coefficients separated by a space. You can also enter full equations with unknowns in the text field (click the example below).

You can add or subtract another equation and coefficients using the buttons or position the cursor over a cell in the last row or column and press the arrow in the appropriate direction on the keyboard.

You can move around cells using tabs, spaces or arrows on the keyboard.

In each field of the calculator you can enter decimal numbers, simple fractions, basic mathematical operation, for example: 1.5; 1/2; 5 + 2; 5-2; 5 * 2; 1/2 + 5; 5 * (8 + 2); (5-2) / 2; 1.5e-3; 1.5 (3); (2/5) * (1/2); 5 + 2 * (5 / 3-2); etc.

You can also enter data into the calculator from the result obtained, just grab any matrix from the results and drop it on cells or a text field.

The results are stored in cookies, which means that the result will be saved until you click .
Thanks to the stored results, you can compare the actions using other linear equation calculators available on our website, eg.

Solving System of linear equations by Cramer's method

Solving Systems with Gauss-Jordan Elimination calculator

System of equations using the inverse matrix method

Example 1:

2x-2y+z=-3
x+3y-2z=1
3x-y-z=2

Example 2:

2x-0y+z=2
x-3y-0z=1
x+y-2z=0

Example 3:

2x-2y+z-w=7
-x+4y-5z+3w=1
x+y-z+w=4
-4x+2y+z-2w=6

Solving Systems with Gaussian Elimination

decimal fractions , number of decimal places =

Result

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Useful information

A system of linear equations is a system of figures:
$$\large{\left\{\begin{array}{1} \mathrm{a_{11}x_{1} + a_{12}x_{2}}&+&\cdots&+&\mathrm{a_{1n}x_{n} = b_{1}},\\ \mathrm{a_{21}x_{1} + a_{22}x_{2}}&+&\dotsb&+&\mathrm{a_{2n}x_{n} = b_{2}},\\ \vdots&\vdots & \ddots & \vdots \\ \mathrm{a_{m1}x_{1} + a_{m2}x_{2}}&+&\cdots&+&\mathrm{a_{mn}x_{n} = b_{m}} \end{array} \right.}$$
gdzie:

$\mathrm{\large{a_{11},...,a_{mn}}}$ - these are the coefficients of the equation (data)
$\mathrm{\large{b_{1},...,b_{m}}}$- free words (data)
$\mathrm{\large{x_{1},...,x_{n}}}$- unknown

There are two types of systems of linear equations:

• homogeneous systems of linear equations,

• systems of linear equations non-homogeneous,
A system of linear equations is called homogeneous if all the terms are zero.
$$\large{\left\{\begin{array}{1} \mathrm{a_{11}x_{1} + a_{12}x_{2}}&+&\cdots&+&\mathrm{a_{1n}x_{n} = 0},\\ \mathrm{a_{21}x_{1} + a_{22}x_{2}}&+&\dotsb&+&\mathrm{a_{2n}x_{n} = 0},\\ \vdots&\vdots & \ddots & \vdots \\ \mathrm{a_{m1}x_{1} + a_{m2}x_{2}}&+&\cdots&+&\mathrm{a_{mn}x_{n} = 0} \end{array} \right.}$$

The methods of solving linear equations are divided into two classes:

• finite methods - allow to obtain a solution after performing a finite number of arithmetic operations, and the obtained solutions are burdened with rounding errors only.

• iterative methods - they consist in determining the sequence of vectors converging to the solution of the system. The obtained solutions are subject to errors in the method and rounding. These methods, however, allow to determine solutions with any predetermined accuracy.

One of the finite methods of solving a system of linear equations is the Gaussian elimination method.

The Gauss elimination method is used to solve systems of equations and consists in reducing the matrix resulting from the equations to the form of a triangular matrix, i.e. to obtain zero under the diagonal (it has been assumed that under the diagonal, however, you can also over the diagonal) of the matrix, making it easier it may also be creating a one on the diagonal, but it is not necessary. To do this, we can swap lines with each other, add them together, and multiply by a number other than zero.

Solving a system of equations using the Gaussian elimination method takes place in two stages:

1. simple procedure stage - the system of equations is transformed into an equivalent form (having the same solution as the initial system) with the triangular upper main matrix of the system,

2. the reverse procedure stage - the agreement in the equivalent form is terminated.

System of linear equations - Wikipedia (EN)
Matrix - Wikipedia (EN)

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