Solve system of linear equations by Cramer's rule online

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

system of linear equations with 3 unknowns

To solve it, one need to find such the values of the variables x1, x2, x3 which convert the initial SLAE to the correct identity. To show how the Cramer's rule works rewrite our initial SLAE in the matrix form:

system of linear equations with 3 unknowns in the matrix form

First step of the Cramer's rule, is to test the value of the determinant of the SLAE matrix:

determinant of the system of linear equations matrix

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

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

determinant of the SLAE does not equal to zero

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

formulas of the Cramer rule

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:

definition of the DeltaX determinant

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 SLAEs where the number of the equations equal to the number of the variables.

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

Solve slae by Cramer's rule
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allow automatic SLE variables detection.
System of linear equations
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