# System of Linear Equations Solver - 2 to 6 Unknowns by Gauss-Jordan

> Free online linear system calculator: fill in the augmented matrix for 2 to 6 unknowns and get the RREF, ranks and the unique, empty or general solution.

- URL: https://qilingtools.com/en/math/linear-system
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Enter one row per equation — the coefficients and then the constant. Gauss-Jordan elimination solves systems of 2 to 6 unknowns and tells you straight away whether the solution is unique, missing or infinite, along with the reduced row echelon form, the ranks and the general solution.

## How to use

1. Choose how many unknowns (2 to 6)
2. Fill in the coefficients row by row, with the constant in the last column
3. Read which case you are in and the value of each unknown
4. If the solution is not unique, read the general solution and the free variables

## Features

- Systems with 2 to 6 unknowns, solved with partial pivoting for numerical stability
- Automatically classifies unique / none / infinitely many, and says why
- Ranks of the coefficient and augmented matrices plus the determinant
- General solution written out with pivot variables in terms of the free ones
- A tidy text version of the system you can copy in one click
- The full reduced row echelon form, so it matches the textbook procedure

## FAQ

### How do I tell whether a system has a solution?

Compare the two ranks. If the augmented matrix has the higher rank there is no solution; if they are equal and match the number of unknowns the solution is unique; if they are equal but smaller there are infinitely many. Both numbers are shown.

### What is the reduced row echelon form?

The augmented matrix after elementary row operations so that each pivot is 1 and every other entry in its column is 0. At that point each row reads off the value of one unknown — the end point of Gauss-Jordan elimination.

### How should I write the answer when there are infinitely many solutions?

Treat every unknown without a pivot as a free variable that can be any real number, and express the rest in terms of them. That is exactly what the “General solution” box shows.

### What does a determinant of 0 mean?

The coefficient matrix is not invertible, so the system either has no solution or infinitely many — which one depends on the rank of the augmented matrix.

### Can I use a different number of equations than unknowns?

The grid is square, but you can enter an all-zero row for a missing equation or repeat one; the rank test still classifies the system correctly, including the no-solution and infinite cases.

## Related tools

- [Graphing Calculator](https://qilingtools.com/en/math/graph): Free online function graphing calculator: type sin(x), x^2 or 1/x and see the curve at once, with several functions, parametric and polar curves, PNG export.
- [Scientific Calculator](https://qilingtools.com/en/math/calculator): Free online scientific calculator: type 2*(3+4)^2, sin(pi/6) or log(8,2) and see the answer as you type, with degrees or radians and a history list.
- [Equation Solver](https://qilingtools.com/en/math/equation): Free online equation solver: enter coefficients for linear, quadratic (with discriminant, vertex and complex roots) and cubic equations, plus 2×2 systems.
- [Fraction Calculator](https://qilingtools.com/en/math/fraction): Free online fraction calculator: add, subtract, multiply and divide fractions with automatic reduction, plus fraction ↔ decimal with repeating digits.
- [Percentage Calculator](https://qilingtools.com/en/math/percentage): Free online percentage calculator: X% of Y, X as a percent of Y, increase and decrease, a value after a percentage change and the original behind it.
- [Prime Factorization](https://qilingtools.com/en/math/prime): Free online prime factorization: factor an integer with exponents, list every divisor and their sum, test primality with neighbouring primes, or list a range.
- [GCD and LCM](https://qilingtools.com/en/math/gcd-lcm): Free online GCD and LCM calculator: works with two or more integers, says whether they are coprime, shows each prime factorization and every Euclidean step.
- [Statistics Calculator](https://qilingtools.com/en/math/statistics): Free online statistics calculator: paste a data set for the mean, median, mode, range, quartiles, variance, standard deviation, standard error and outliers.