System of Linear Equations Solver (Gaussian Elimination)

Enter the augmented matrix [A|b] and this tool solves the system of linear equations Ax=b by Gauss-Jordan elimination. It detects a unique solution, no solution (inconsistent), or infinitely many solutions, and shows the solution vector and reduced row echelon form.

Input

Enter the augmented matrix [A|b]. One equation per row; separate numbers with spaces or commas. The last number on each row is the right-hand-side constant.

Augmented matrix [A|b]

Result

Solution

x = 2

y = 3

z = -1

Variables

Equations

Solution type

Unique solution

Rank-based check

rank(A) = 3, rank([A|b]) = 3

Reduced row echelon form (RREF)

The augmented matrix after Gauss-Jordan elimination. The rightmost column corresponds to the constant term b.

1	0	0	2
0	1	0	3
0	0	1	-1

How it works

Enter the augmented matrix [A|b]. Each row is one equation, and the numbers in a row are separated by spaces or commas. The last number on each row is the right-hand-side constant b.
Example: 2x+y−z=8 is entered as "2 1 -1 8". Include zero coefficients explicitly as 0; do not omit them.
The tool reduces the matrix to reduced row echelon form (RREF) using Gauss-Jordan elimination, with partial pivoting (choosing the row with the largest absolute pivot) for numerical stability.
The solution type is determined by rank: if rank(A) = rank([A|b]) = number of variables there is a unique solution; if they are equal but smaller there are infinitely many solutions; if they differ the system has no solution (inconsistent).
Because the computation uses floating-point arithmetic, very small values are rounded to 0 for display. Even when an integer solution is expected, tiny rounding remainders may appear.

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