Linear System Solver

Math Linear Algebra • Matrices and Systems of Equations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Solve a linear system \(A\mathbf{x}=\mathbf{b}\) using row-reduction of the augmented matrix \([A\mid \mathbf{b}]\). The solver classifies the system as having a unique solution, infinitely many solutions, or no solution.

Equations (m)

Variables (n)

Display decimals

Pivot tolerance

Steps

Input the augmented matrix \([A\mid b]\). The last column is \(\mathbf{b}\). Play animates row-reduction snapshots once and stops automatically.

Augmented matrix \([A\mid b]\)

2×(2+1)

RREF \([A\mid b]\)

—

Ready

Row operation animation (augmented matrix)

Drag to pan • wheel to zoom • double-click to reset. Pivot cell and pivot row are highlighted during Play.

A block b column pivot pivot row eliminated row

Enter a system and click “Calculate”.

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Theory

Linear systems and the matrix form Ax=b

A linear system is a collection of linear equations in one or more unknowns. In matrix form, many systems can be written as \[ A\mathbf{x}=\mathbf{b}, \] where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the vector of unknowns, and \(\mathbf{b}\) is the constant vector. If there are \(m\) equations and \(n\) unknowns, then \(A\) is an \(m\times n\) matrix, \(\mathbf{x}\in\mathbb{R}^n\), and \(\mathbf{b}\in\mathbb{R}^m\).

The most common tool for solving such systems is row reduction. Instead of manipulating equations one at a time, we combine everything into an augmented matrix: \[ [A\mid \mathbf{b}], \] where the last column stores the constants. Each row corresponds to an equation, and each column (except the last) corresponds to a variable.

Row operations and why they preserve solutions

Gaussian elimination and Gauss–Jordan elimination use three elementary row operations:

Swap two rows.
Multiply a row by a nonzero constant.
Add a multiple of one row to another row.

These operations do not change the set of solutions of the system (they produce an equivalent system). In practical terms, a row swap just reorders equations, scaling a row multiplies an equation by a nonzero constant, and adding a multiple of one row to another replaces an equation by a linear combination of equations that is still logically implied by the original system.

Gauss–Jordan elimination continues until the matrix is in reduced row echelon form (RREF). In RREF, each pivot column contains a leading 1 (the pivot) and zeros everywhere else in that column. This makes the solution structure very easy to read directly from the augmented matrix.

Unique, infinite, and no solutions

After row reduction, a system falls into one of three categories:

No solution (inconsistent): a row like \([0\ 0\ \cdots\ 0\mid c]\) with \(c\neq 0\) appears. This represents an impossible equation such as \(0=5\).
Unique solution: every variable column is a pivot column. Then the RREF effectively becomes \([I\mid \mathbf{x}]\), so each variable is determined by a single number.
Infinitely many solutions: the system is consistent, but at least one variable column has no pivot. Those variables are called free variables, and the solution set is described with parameters (for example \(t_1,t_2,\dots\)).

The number of pivots corresponds to the rank of the coefficient matrix. When there are fewer pivots than variables, at least one degree of freedom remains, producing infinitely many solutions.

Application tease: circuit equations

Many engineering and physics applications naturally produce linear systems. In circuit analysis, Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) produce equations relating currents and voltages. Once written in standard form, these become \(A\mathbf{x}=\mathbf{b}\) with unknown node voltages or branch currents. Solving the system is exactly the same row-reduction process you practice in algebra—only the interpretation changes.

The animation in the solver highlights pivot choices and row operations, which is a visual version of the elimination steps often shown in textbook circuit examples. This helps connect the abstract matrix procedure to the practical workflow used in engineering.

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