Gaussian Elimination Calculator
Gaussian elimination is the workhorse method for solving systems of linear equations. It uses elementary row operations to reduce the augmented matrix to upper triangular form, then sweeps back up through the rows to recover each unknown. This calculator solves a 3x3 system using partial pivoting for numerical stability, choosing the largest available pivot in each column. Enter each equation as its three coefficients and its constant. The tool returns x, y, and z, or flags a system with no unique solution.
Gaussian elimination steps
1. Form the augmented matrix [A | b]
2. For each column pick the largest pivot row
3. Eliminate entries below the pivot
4. Reach upper triangular form
5. Back substitute from the bottom up
Row operations preserve the solution set. After reaching upper triangular form, the last equation gives one unknown, which is substituted upward to find the rest.
Notes on Gaussian elimination
- Each equation is four comma-separated numbers: three coefficients and a constant.
- Partial pivoting selects the largest available coefficient as each pivot.
- A zero pivot after elimination signals no unique solution.
- The method scales far better than Cramer's rule for large systems.
- Substitute the solution back into each equation to verify.
Gaussian elimination: frequently asked questions
What is Gaussian elimination?
Gaussian elimination is a systematic method for solving a system of linear equations. It uses elementary row operations to transform the augmented matrix into upper triangular form, then solves for the unknowns by back substitution from the bottom row upward.
What is partial pivoting?
Partial pivoting swaps rows so that the largest available coefficient in each column becomes the pivot. This reduces the chance of dividing by a tiny number, which improves numerical stability and accuracy of the solution.
What is back substitution?
Once the matrix is upper triangular, the last equation gives one unknown directly. That value is substituted into the equation above to solve the next unknown, and the process continues upward until all unknowns are found.
When does Gaussian elimination have no unique solution?
If, after elimination, a pivot position contains zero, the system is either inconsistent (no solution) or dependent (infinitely many solutions). In either case there is no unique solution and this calculator returns n/a.
How do I enter my system?
Each equation is one row of four comma-separated numbers: the three coefficients followed by the right-hand side constant. For example 2, 1, -1, 8 represents the equation 2x + y - z = 8.
Official sources
- NIST Digital Library of Mathematical Functions: Determinants and linear operators.
- NIST/SEMATECH e-Handbook of Statistical Methods: Linear algebra reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.