System Of Linear Equations Calculator

Work out system of linear equations instantly with clear inputs, formula shown and shareable results.

About System Of Linear Equations Calculator

Gauss-Jordan elimination with partial pivoting reduces the augmented matrix to the identity, leaving the solution in the final column. Partial pivoting keeps the arithmetic numerically stable.

Formula

Gauss-Jordan

reduce [A | b] to [I | x] by row scaling and row subtraction, pivoting on the largest available entry

Worked example

With the default values below, System Of Linear Equations Calculator returns Solution of x1 = 1, x2 = 2, x3 = 3. Change any input above and every figure here is recalculated in your browser.

Inputs used

First equation coefficients then constant: 1, 2, 3, 14
Second equation: 2, -1, 1, 3
Third equation: 3, 1, -2, -1
Number of unknownsone of Two, Three: Three

Results

Solution: x1 = 1, x2 = 2, x3 = 3
First unknown: 1.00000000
Second unknown: 2.00000000
Third unknown: 3.00000000

Step by step

1Values used
2Gauss-Jordan
3Solution= x1 = 1, x2 = 2, x3 = 3
4First unknown= 1.00000000
5Second unknown= 2.00000000
6Third unknown= 3.00000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Solve x + 2y + 3z = 14, 2x − y + z = 3, 3x + y − 2z = −1.

The reduction gives x = 1, y = 2, z = 3.

Why pivot on the largest entry?

Dividing by a tiny pivot amplifies rounding error; choosing the largest available magnitude keeps the computation stable.