Can I enter full equations?

This page accepts prepared coefficients. Rearrange each equation first, then enter the coefficient matrix.

What does a zero main determinant mean?

Cramer's rule cannot produce a unique solution. Gaussian elimination is needed to classify the system.

How is Gaussian elimination different from Cramer's rule?

Cramer's rule uses determinants, while Gaussian elimination uses row operations on the augmented matrix.

Can I enter fractions?

Yes. Use in any cell. The calculation is numeric, so long decimal results may be rounded.

What if a variable is missing from one equation?

Enter 0 for that variable's coefficient so the row still has x, y, z, and the constant.

Why is there no single triple for infinitely many solutions?

A system with free variables has a family of solutions, not one ordered triple. This tool classifies the case but does not output the full parametric form.

3x3 System of Equations Solver

Solve a 3x3 system of linear equations by coefficients with Cramer's rule, Gaussian elimination steps, classification, and answer check.

Enter coefficients for three equations.

x, row 1

y, row 1

z, row 1

constant 1

x, row 2

y, row 2

z, row 2

constant 2

x, row 3

y, row 3

z, row 3

constant 3

Virtual keypad for the active cell

Examples

Solution

Main determinant

x determinant

y determinant

z determinant

Method values

The calculator computed the main determinant and three replacement determinants. If the main determinant is nonzero, the system has one solution.

Substitution checks the values against each original row.

Gaussian elimination check

The augmented matrix is reduced to row-echelon form to confirm the same solution type.

A 3x3 linear system is solved from three coefficient rows. The calculator shows variable values, determinant checks, and Gaussian elimination classification.

What the 3x3 system solver calculates

This page uses coefficient input. Fill three rows of a 3x4 augmented matrix: coefficients of x, y, z, and the constant for each equation. It is the fast workflow when the system is already in linear form.

⎩ ⎨ ⎧ a_{11} x + a_{12} y + a_{13} z = b_{1} a_{21} x + a_{22} y + a_{23} z = b_{2} a_{31} x + a_{32} y + a_{33} z = b_{3}

x, y, and z are the unknowns; a11 through a33 are coefficients; b1 through b3 are constants.

Format	What to enter	Best when
Coefficient matrix	three coefficients and one constant per row	the coefficients are already known
Free-form equation text	three complete equations	the equations still need parsing or rearranging

If you have full equations

Move each equation into standard linear form before entering coefficients. This page does not parse full equation text.

Cramer's rule and determinants

Cramer's rule calculates the main determinant of the coefficient matrix and three replacement determinants. If the main determinant is nonzero, the system has one ordered triple.

x = \frac{Δ _{x}}{Δ}, y = \frac{Δ _{y}}{Δ}, z = \frac{Δ _{z}}{Δ}

The replacement determinants are divided by the main determinant Delta.

Cramer's rule limit

When the main determinant is zero, Cramer's rule does not produce a unique answer. The calculator also shows Gaussian elimination steps to classify the case.

Gaussian elimination

Gaussian elimination works on the augmented matrix. Forward elimination clears entries below pivot positions, and back substitution finds the values when a unique solution exists. The row-echelon form also reveals contradictions and free variables.

Enter the three coefficient rows and constants
Review the main and replacement determinants
Compare the Cramer's rule result with Gaussian elimination
Open the substitution check for a unique solution

Three result types

Signal	Calculator output	Meaning
Main determinant is nonzero	values of x, y, z and a check	unique solution
Main determinant is zero with no contradiction	infinitely many solutions	free variables exist
Gaussian elimination finds a contradictory row	no solution	inconsistent system

No parametric family

For infinitely many solutions, this calculator classifies the system but does not output a full parametric solution set.

Cramer's rule vs Gaussian elimination

Criteria	Cramer's rule	Gaussian elimination
Uses	determinants	row operations
Numerical answer	when the main determinant is nonzero	after forward elimination and back substitution
Visible steps	main and replacement determinants	augmented matrix and row-echelon form
Main value	compact answer check	classification of singular cases

How to prepare the system

Move the variable terms to the left and leave the constant on the right. If a variable is missing in a row, enter zero in that coefficient cell. Fractions can be entered as.

Step	Example
Original row	x equals y plus z plus one
After moving terms	coefficients of x, y, z are 1, -1, -1
Matrix input	enter 1, -1, -1, 1

Frequently Asked Questions

Sources and References

System of Equations
Wolfram MathWorld

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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