Build and inspect systems with responsive matrix inputs. View steps, residuals, ranks, and determinant summaries. Download polished reports and compare scenarios through interactive charts.
Designed for simultaneous linear equation systemsBuild a system from 2 to 6 equations and 2 to 6 variables. This solver handles linear systems written in matrix form.
This sample system has three variables and one exact solution. It matches the classic preset above.
| Equation | x1 coefficient | x2 coefficient | x3 coefficient | Constant |
|---|---|---|---|---|
| 2x1 + x2 - x3 = 8 | 2 | 1 | -1 | 8 |
| -3x1 - x2 + 2x3 = -11 | -3 | -1 | 2 | -11 |
| -2x1 + x2 + 2x3 = -3 | -2 | 1 | 2 | -3 |
Expected solution: x1 = 2, x2 = 3, x3 = -1.
The calculator models the system as A x = b, where A is the coefficient matrix, x is the variable vector, and b is the constants vector.
For exact solutions, it converts the augmented matrix [A|b] into reduced row-echelon form using Gauss-Jordan elimination. The rank rules are:
When approximation mode is enabled and an exact answer is unavailable, the fitted estimate uses the normal equations: x = (ATA)-1ATb.
It solves simultaneous linear systems in matrix form. Each equation should contain the same variables with numeric coefficients and one constant term.
Yes. Overdetermined systems are supported. If the equations remain consistent, you can still get one exact answer. If they conflict, approximation mode can return a least-squares fit.
For square systems, a nonzero determinant usually indicates a unique solution. A determinant close to zero warns that the system may be unstable or nearly singular.
Residuals measure the difference between the observed constants and the values reconstructed from the reported solution. Smaller residuals mean the answer fits the system more closely.
That pattern means the system is inconsistent. At least one equation contradicts the others, so no exact solution satisfies the entire set simultaneously.
Turn it on when you want a best-fit estimate for inconsistent or noisy systems, especially when your equations come from measurements, experiments, or rounded values.
Yes. Check the elimination steps option before solving. The calculator will display row swaps, normalization steps, and elimination stages used to build the final reduced matrix.
Yes. After solving, you can download a CSV summary for spreadsheet work or a PDF snapshot for sharing, printing, or attaching to documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.