Linear Equation Solver

Solve systems of linear equations¹ \(A\mathbf{x} = \mathbf{b}\) using row reduction² with detailed step-by-step output.
Handles consistent, inconsistent, and underdetermined systems.

A system of linear equations \(A\mathbf{x} = \mathbf{b}\) can be:

Consistent with unique solution: Exactly one solution exists
Consistent with infinitely many solutions: Free variables present (underdetermined)
Inconsistent: No solution exists (contradictory equations)³

This solver uses Gauss-Jordan elimination⁴ to reduce the augmented matrix \([A|\mathbf{b}]\) to reduced row echelon form (RREF) and determine the solution set.

Equations (rows, \(m\)): Variables (columns, \(n\)):

Enter dimensions and click Generate Matrix to create the augmented input grid.

Display values as: Decimal Fraction

1 — Compute

Solves the linear system using step-by-step row reduction and back substitution.

Resets all dimensions, matrix entries, and results.

Footnotes

A system of linear equations \(A\mathbf{x} = \mathbf{b}\) consists of \(m\) equations in \(n\) unknowns, where \(A\) is the \(m \times n\) coefficient matrix, \(\mathbf{x}\) is the vector of unknowns, and \(\mathbf{b}\) is the right-hand side vector. ↑
Row reduction (Gaussian elimination) applies elementary row operations to transform the augmented matrix \([A|\mathbf{b}]\) into an equivalent system that is easier to solve. ↑
An inconsistent system arises when the RREF contains a row of the form \([0 \cdots 0 \mid c]\) with \(c \neq 0\), meaning the equations are contradictory. ↑
Gauss-Jordan elimination extends Gaussian elimination by continuing to reduce the matrix to reduced row echelon form (RREF), where each pivot is 1 and is the only nonzero entry in its column. This allows the solution to be read off directly. ↑

Cite this tool

DOI: https://doi.org/10.5281/zenodo.20981279

Kapita, S. (2026). Linear Equation Solver. Math Tools. https://doi.org/10.5281/zenodo.20981279

Kapita, Shelvean. "Linear Equation Solver." Math Tools, 2026, doi.org/10.5281/zenodo.20981279.

@online{kapita2026linearequation,
  author       = {Shelvean Kapita},
  title        = {{Linear Equation Solver}},
  year         = {2026},
  organization = {Math Tools},
  doi          = {10.5281/zenodo.20981279},
  url          = {https://doi.org/10.5281/zenodo.20981279}
}