Built and fact-checked by the DocNectar team — see our editorial standards
Key Features
Instant Calculation
Get accurate results in real time with our optimized algorithm.
Mobile Friendly
Fully responsive design. Works on all devices & screen sizes.
Privacy Focused
Your data stays on your device. We don't store any inputs.
100% Free
No hidden costs. This tool is completely free forever.
Gauss-Jordan elimination solves a system of linear equations by fully reducing it to a form where the answer can simply be read off, rather than computing determinants like Cramer's rule — and unlike Cramer's rule, it scales well to larger systems and to systems that aren't square.
This calculator solves a system with any number of equations (rows) and unknowns (columns), each up to 6, using Gauss-Jordan elimination — including systems with more equations than unknowns or more unknowns than equations.
How it works
Enter the number of equations and unknowns, then the coefficient matrix and the constants on the right-hand side. The calculator combines them into one augmented matrix, fully reduces it, and classifies the result as a unique solution, infinitely many solutions (when there are free variables), or no solution (when the equations are inconsistent).
- Click Calculate to see your results.
Examples
A 2×2 system
For 2x + y = 8 and x + 3y = 13, Gauss-Jordan elimination gives x = 2.2 and y = 3.6 — the same answer Cramer's rule gives, reached a different way.
A 3-equation, 2-unknown system
A system like x + y = 3, x − y = 1, and 2x = 4 has 3 equations but only 2 unknowns — Gauss-Jordan elimination confirms all three are consistent and gives the unique solution x = 2, y = 1.
Who should use it
- Solving systems of linear equations for coursework.
- Learning the standard elimination method taught in most linear algebra courses.
Industry applications
- Linear algebra and mathematics education
- Engineering, physics, and applied mathematics
Advantages
- Scales much better to larger systems than determinant-based methods.
- Shows the full augmented matrix and its reduced form, not just the final answer.
Limitations
- Involves more computational steps to follow by hand than simply plugging into a formula for very small systems.
Common mistakes to avoid
- Stopping at row echelon form instead of continuing to reduced row echelon form — Gauss-Jordan specifically continues until the left side is the identity matrix.
- Mixing up which column holds the constants versus the coefficients when building the augmented matrix.
Best practices
- Work through the elimination systematically, one pivot column at a time, and use the reduced row echelon form shown in the steps to check your own hand-worked solution.
Tips
- Gauss-Jordan elimination is the same underlying method used to compute a matrix inverse by hand — reducing [A | I] to [I | A⁻¹] instead of [A | b] to [I | x].