This professional system of equations calculator solves linear systems from 2×2 up to 5×N using numerically stable Gaussian elimination. It reveals every step toward the Reduced Row Echelon Form (RREF), classifies the system (unique, infinite, none), and, for square systems, computes the determinant—all within an accessible, mobile-first interface.
Data Source and Methodology
Authoritative source: Gilbert Strang, “Introduction to Linear Algebra,” 5th Edition, Wellesley-Cambridge Press, 2016. MIT OpenCourseWare companion materials: 18.06 Linear Algebra Lecture Notes. Published 2010; textbook 2016.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
We solve the linear system using the augmented matrix and elementary row operations.
System: $$A\\mathbf{x} = \\mathbf{b},\\quad A \\in \\mathbb{R}^{m\\times n}$$
Augmented matrix: $$[A\\,|\\,\\mathbf{b}]$$
Row operations (preserve solution set):
1) Swap rows: $$R_i \\leftrightarrow R_j$$ 2) Scale a row: $$R_i \\leftarrow c\\,R_i,\\ c\\neq 0$$ 3) Row replacement: $$R_i \\leftarrow R_i - k R_j$$
Reduced Row Echelon Form: leading 1s with zeros above and below. If the system is square and invertible, the unique solution is
$$\\mathbf{x} = A^{-1}\\mathbf{b}$$
Determinant (square case) from elimination: $$\\det(A) = (-1)^{s}\\,\\prod_{i=1}^{n} u_{ii}$$ where s is the number of row swaps and uii are pivots prior to normalization.
Glossary of Variables
- aij: Coefficient at row i, column j in A.
- bi: Constant term at row i.
- xj: Unknown variable j.
- [A|b]: Augmented matrix used for elimination.
- Rank(A): Number of pivots in A; rank([A|b]) includes the augmented column.
- RREF: Canonical form achieved by Gaussian elimination and back substitution.
- Determinant: Scalar indicating invertibility for square matrices (nonzero ⇔ unique solution).
Come Funziona: Un Esempio Passo-Passo
Consider the 3×3 system
Augmented matrix:
After Gaussian elimination to RREF, we obtain
The calculator replicates these steps, showing row swaps, scalings, and eliminations transparently. Because the determinant is nonzero, the solution is unique.
Frequently Asked Questions (FAQ)
What kinds of systems can I enter?
Any real-valued linear system with 2–5 equations and up to 5 variables. Rectangular systems are allowed; the tool classifies them accordingly.
How does the tool detect no or infinite solutions?
It computes the RREF of the augmented matrix. A row like [0 0 … 0 | c≠0] indicates inconsistency (no solution). Rank(A) = Rank([A|b]) < number of variables indicates infinitely many solutions with free variables.
Are blank cells allowed?
Yes. A blank cell is treated as 0. You can also use scientific notation, e.g., 1e-3.
How accurate is the method?
Gaussian elimination with partial pivoting is standard and reliable for small systems. A tolerance is applied to handle near-zero pivots.
Can I rename variables?
Yes. Provide comma-separated labels like x,y,z or a,b,c,d. Otherwise, the tool uses x₁, x₂, x₃, …
Do you show the determinant?
Yes, for square systems. A nonzero determinant implies a unique solution; zero implies either no or infinitely many solutions.
Formula (LaTeX) + variables + units
A\\mathbf{x} = \\mathbf{b},\\quad A \\in \\mathbb{R}^{m\\times n}
[A\\,|\\,\\mathbf{b}]
R_i \\leftrightarrow R_j
R_i \\leftarrow c\\,R_i,\\ c\\neq 0
R_i \\leftarrow R_i - k R_j
\\mathbf{x} = A^{-1}\\mathbf{b}
We solve the linear system using the augmented matrix and elementary row operations. System: $A\\mathbf{x} = \\mathbf{b},\\quad A \\in \\mathbb{R}^{m\\times n}$ Augmented matrix: $[A\\,|\\,\\mathbf{b}]$ Row operations (preserve solution set): 1) Swap rows: $R_i \\leftrightarrow R_j$ 2) Scale a row: $R_i \\leftarrow c\\,R_i,\\ c\\neq 0$ 3) Row replacement: $R_i \\leftarrow R_i - k R_j$ Reduced Row Echelon Form: leading 1s with zeros above and below. If the system is square and invertible, the unique solution is $\\mathbf{x} = A^{-1}\\mathbf{b}$ Determinant (square case) from elimination: $\\det(A) = (-1)^{s}\\,\\prod_{i=1}^{n} u_{ii}$ where s is the number of row swaps and uii are pivots prior to normalization.
$\\begin{cases} 2x + y - z = 8 \\\\ -3x - y + 2z = -11 \\\\ -2x + y + 2z = -3 \\end{cases}$
$\\left[\\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & 3 \\\\ 0 & 0 & 1 & -1 \\end{array}\\right],\\quad \\Rightarrow\\ (x,y,z)=(2,3,-1)$
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- Math — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/math - 18.06 Linear Algebra Lecture Notes — ocw.mit.edu · Accessed 2026-01-19
https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/pages/lecture-notes/
Last code update: 2026-01-19
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