Linear Equation Solver

Solve systems of linear equations with step-by-step Gaussian elimination, interactive augmented matrix visualization, and exact fraction arithmetic.

System Size (2 × 2):

Display

Equations

Augmented Matrix [A|b]

x	y	b

Solution

Infinitely Many Solutions

Free variables x, y

x = t_{1} \quad \text{(free)}

y = t_{2} \quad \text{(free)}

0 = 0

0 = 0

Augmented Matrix

Original System

[	$0$	$0$	$0$	]
[	$0$	$0$	$0$	]

0 / 0

Open this tool with your own values

This tool reads its inputs from the page address, so a link can open it with the values from a problem already entered. Include every value the problem needs, because one left out keeps the tool's default. Use plain numbers in the stated units.

n: number of equations and unknowns, 1 to 10
m: coefficients then the constant of each equation, equation by equation, comma-separated, as whole numbers or fractions written like 1_2

Example link

https://livephysics.com/tools/linear-equation-solver/?n=3&m=1,1,1,6,2,-1,1,3,1,2,-1,2

Reference Guide

Types of Solutions

Unique Solution Lines (or planes) intersect at exactly one point. The system has full rank.

No Solution Lines are parallel (inconsistent). At least one row reduces to

0 = c

where

c \neq 0

.

Infinite Solutions Lines overlap (dependent). Free variables appear, described in parametric form.

Gaussian Elimination

A systematic algorithm to solve systems by transforming the augmented matrix to reduced row echelon form (RREF).

Forward elimination Use row operations to create zeros below each pivot
Scale pivots Divide each pivot row so the pivot becomes 1
Back substitution Eliminate above each pivot to get RREF

\begin{bmatrix} 1 & 1 & 3 \\ 1 & -1 & 1 \end{bmatrix} \xrightarrow{R_2 - R_1} \begin{bmatrix} 1 & 1 & 3 \\ 0 & -2 & -2 \end{bmatrix}

Cramer's Rule

For a 2×2 system $Ax = b$ , when $\det(A) \neq 0$ :

x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}

Where $A_x$ replaces column 1 of $A$ with $b$ , and similarly for $A_y$ .

For a 2×2 matrix $\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

Matrix Rank & Solution Type

The rank of a matrix is the number of non-zero rows after row reduction (the number of pivot positions).

$\text{rank}(A) = \text{rank}([A|b]) = n$ : Unique solution

$\text{rank}(A) < \text{rank}([A|b])$ : No solution

$\text{rank}(A) = \text{rank}([A|b]) < n$ : Infinite solutions with $n - \text{rank}(A)$ free variables

TopicsAlgebra Algorithms Fractions Process Mathematics

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