How this calculator works
Enter one row per equation in row order, placing the constant after the coefficients. The algorithm scales non-zero coefficient rows, swaps for a large pivot, makes that pivot one, and eliminates its column above and below. A contradictory zero-coefficient row means no solution. Otherwise, a pivot for every variable gives a unique solution; missing pivots identify free variables. The relative threshold is 1e-12 after row scaling, so nearly dependent systems need higher-precision analysis.
Formula
For Ax = b, elementary row operations preserve solutions. Reduce [A|b] to pivots of one with zero entries above and below. A row [0 ... 0 | c], c != 0, is inconsistent.
Worked example
The equations x + y = 3 and x - y = 1 reduce to x = 2 and y = 1. For 2y = 4 and x = 3, a row swap gives x = 3 and y = 2.
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Frequently asked questions
How is this different from Gaussian elimination?
Gauss-Jordan elimination also clears entries above each pivot, so a unique solution can be read directly without back substitution.
What does a free variable mean?
A non-pivot variable can vary. Use the reduced rows to express pivot variables in terms of the free variables.
Is the result exact?
The method uses floating-point arithmetic with scaled pivoting. Ill-conditioned inputs can change rank decisions near the stated threshold.