# Solve system of linear equations by Cramer's rule online

One of the ways to solve the system of the linear algebraic equations (SLE) is to use the *Cramer's rule*.
Suppose we have the SLE:

To solve it, one need to find such the values of the variables
x_{1},
x_{2},
x_{3}
which convert the initial SLE to the correct identity. To show how the Cramer's rule works rewrite our initial SLE in the matrix form:

First step of the *Cramer's rule* is to test the value of the determinant of the SLE matrix:

If the computed determinant doesn't equal to zero, then the initial SLE has the single solution, which can be found by the Cramer's rule. If the computed determinant does equal to zero, then the initial SLE may either have no solution or have the inifinity set of the solutions which cannot be found by the Cramer's rule.

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Suppose, the computed determinant doesn't equal to zero:

then, by the Cramer's rule, the solution of the SLE can be found by using the formulas:

here, ∆_{x}, ∆_{y} and ∆_{z} are the
determinants derived from the determinant ∆ by changing the corresponding column to the vector of the free coefficients.
For example, the determinant ∆_{x} received from the determinant ∆
by changing the first column to the vector of the free coefficients:

By using this method one can receive the determinants ∆_{y} and ∆_{z}.
It should be noted, that the Cramer's rule is applicable to the SLE's where the number of the equations equal to the number of the variables.

Our online calculator *solves the SLE by the Cramer's rule* with step by step solution. SLE coefficients can be not only numbers of fractions but also the parameters.
To use the calculator, one need to input the SLE and choose the SLE's variables to find.

## Choose expression input type:

## Choose number of equations in system and the variables:

Number of equations in system:

Variables you want to find: