How this calculator works
Gauss-Jordan elimination scales rows, chooses the largest available pivot and clears each pivot column. A contradictory row gives no solution. Fewer pivots than unknowns, without contradiction, gives infinitely many solutions. Otherwise the right column contains the unique solution. Relative pivot tolerance is 1e-12; nearly dependent systems may be classified as singular.
Formula
For A*x = b, row operations on [A | b] preserve the solution set. Unique solutions reduce the coefficient block to the identity matrix.
Worked example
The rows 1, 1, 5 and 1, -1, 1 mean x + y = 5 and x - y = 1, giving x = 3 and y = 2. The three equations x+y+z=6, x-y+z=2 and 2x+y-z=1 give (1, 2, 3).
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Frequently asked questions
Can I enter nonlinear equations?
No. Each row must be linear in x, y and optionally z. Products such as xy need a different solver.
What if the equations repeat?
Repeated equations do not add an independent constraint. The solver reports infinitely many solutions unless another row creates a contradiction.
Why can nearly dependent systems be unreliable?
Small input changes can cause large changes in the solution. This numerical method uses a tolerance and is not an exact symbolic rank calculation.