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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]

xyb

Solution

Infinitely Many Solutions

Free variables x, y

x=t1(free)x = t_{1} \quad \text{(free)}
y=t2(free)y = t_{2} \quad \text{(free)}
0 = 0
0 = 0

Augmented Matrix

Original System
[000000]
000000
0 / 0

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=c0 = c where c≠0c \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).

  1. Forward elimination Use row operations to create zeros below each pivot
  2. Scale pivots Divide each pivot row so the pivot becomes 1
  3. Back substitution Eliminate above each pivot to get RREF
[1131−11]→R2−R1[1130−2−2]\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=bAx = b, when det⁡(A)≠0\det(A) \neq 0:

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

Where AxA_x replaces column 1 of AA with bb, and similarly for AyA_y.

For a 2×2 matrix: det⁡(abcd)=ad−bc\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).

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

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

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