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.

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

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.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

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How to use System of Equations

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

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.