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Solving any systems with Cramer's Rule calculator

You can calculate step by step any system of linear equations, both homogeneous and inhomogeneous with any number of unknowns by the Cramer's method. As a result, in addition to the solution, you also get a complete analysis and presentation of calculations step by step.
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 Systems with Gaussian Elimination calculator

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

System of linear equations by Cramer's method

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.

Cramer's theorem:
If the determinant of the matrix of coefficients of the system is different from zero ( detA≠ 0 ), then the system of linear equations in the vector form $ \color{blue}{ x_1} \mathbf a_1 + \dots + \color{blue}{x_n} \mathbf a_n = \color{red}{ \mathbf b}$ has exactly one solution given by formulas:
$$\color{blue}{x_1} = \frac{\det(\color{red}{\mathbf b},\; \mathbf a_2, \dots, \mathbf a_n)}{\det(\mathbf a_1, \mathbf a_2, \dots, \mathbf a_n)},$$ $$\vdots$$ $$\color{blue}{x_n} = \frac{\det(\mathbf a_1, \dots, \mathbf a_{n-1},\; \color{red}{ \mathbf b})}{\det(\mathbf a_1, \dots, \mathbf a_{n-1}, \mathbf a_n)}$$. so
$$x_i = \frac{W_i}{W};$$where

$W = detA$ - it is the determinant of the matrix of coefficients of the system. This is called the main determinant.

$W_i$ - it is the determinant of the matrix that is derived from the matrix $A $, by replacing the column of the coefficients of the unknown $\ color{blue}{x_i} $ with the column of intercepts $\ color{red}{b} $.

The system of equations can be written in the matrix form:
$$\begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} \begin{bmatrix} \color{blue}{x_1} \\ \color{blue}{x_2} \\ \vdots \\ \color{blue}{x_n} \end{bmatrix} = \begin{bmatrix} \color{red}{b_1} \\ \color{red}{b_2} \\ \vdots \\ \color{red}{b_n} \end{bmatrix} .$$ Given a square matrix of intercepts, where the number of lines is equal to the number of columns (m = n), let's calculate the main determinant W.
$$W = \begin{vmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{vmatrix}$$ If W ≠ 0 we compute the determinants of auxiliary matrices W ₁, W ₂ ... W _n replacing the corresponding coefficient column with the intercept column.
$$W_1 = \begin{vmatrix} \color{red}{b_1} & a_{12} & \dots & a_{1n} \\ \color{red}{b_2} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ \color{red}{b_n} & a_{m2} & \dots & a_{mn} \end{vmatrix}; W_2 = \begin{vmatrix} a_{11} & \color{red}{b_1} & \dots & a_{1n} \\ a_{21} & \color{red}{b_2} & \dots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & \color{red}{b_n} & \dots & a_{mn} \end{vmatrix}; ...; W_n = \begin{vmatrix} a_{11} & a_{12} & \dots & \color{red}{b_1} \\ a_{21} & a_{22} & \dots & \color{red}{b_2} \\ \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \dots & \color{red}{b_n} \end{vmatrix}$$
Now, according to the above formula, substituting determinants, we can calculate the unknowns.

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

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