Gaussian Elimination Calculator
Free Gaussian elimination calculator with step-by-step row operations. Solve systems of linear equations, find row echelon form, and read off the solution with exact fractions.
Enter an augmented matrix and watch every row operation that takes it to echelon form. The calculator names each operation, shows the matrix after every step, and reports whether the system has one solution, none, or infinitely many.
How to solve a system with Gaussian elimination
- 1
Write the augmented matrix
One row per equation, one column per coefficient, and a last column for the constants.
- 2
Create the pivots
Swap rows if needed so the pivot is non-zero, then scale that row so the pivot is 1.
- 3
Clear below each pivot
Subtract multiples of the pivot row from the rows underneath until the column has zeros below the pivot.
- 4
Back-substitute
Work upward from the last equation, substituting known values until every variable is found.
The worked example above
The matrix loaded into the calculator represents this system:
2x + y − z = 8 −3x − y + 2z = −11 −2x + y + 2z = −3
Elimination reduces it to x = 2, y = 3, z = −1. Substituting back into the first equation gives 2(2) + 3 − (−1) = 8, which confirms the answer.
Three possible outcomes
Exactly one solution
Every variable column has a pivot. The rank equals the number of unknowns, and the lines, planes or hyperplanes meet at a single point.
No solution
A row reduces to 0 = c with c non-zero. The equations contradict each other, so nothing satisfies them all.
Infinitely many
At least one column has no pivot, so its variable is free. The solution is a line, plane or larger family described in terms of those free variables.
Gaussian elimination by system size
Frequently asked questions
What is Gaussian elimination? ▾
Gaussian elimination is a systematic way of solving a linear system by using row operations to create zeros below each pivot, turning the augmented matrix into row echelon form. Once the matrix is triangular, back-substitution gives the values of the variables.
What is the difference between Gaussian elimination and Gauss-Jordan elimination? ▾
Gaussian elimination stops at row echelon form — zeros below the pivots — and then uses back-substitution. Gauss-Jordan keeps going until the matrix is in reduced row echelon form, with zeros above the pivots too and 1s on the diagonal, so the solution can be read straight off.
What are the three elementary row operations? ▾
Swapping two rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another. All three preserve the solution set, which is why the method is valid.
How do you know if a system has no solution? ▾
If elimination produces a row where every coefficient is zero but the constant is not — that is, 0 = 5 — the system is inconsistent and has no solution. The calculator flags this for you.
What is a free variable? ▾
A variable whose column contains no pivot. Free variables can take any value, and the pivot variables are then expressed in terms of them, which gives infinitely many solutions.
How do I enter a system of equations? ▾
Enter the augmented matrix: one row per equation, one column per variable, and a final column holding the constants on the right-hand side. Tick the box below the grid to have the solution read out for you.
About this Calculator
Free Gaussian elimination calculator with step-by-step row operations. Solve systems of linear equations, find row echelon form, and read off the solution with exact fractions.