Enter coefficients, constants, and precision for reliable system solving. View matrices, determinants, and elimination steps. Export results, inspect graphs, and verify every variable carefully.
Enter four linear equations in the form ax + by + cz + dw = e. The interface uses a three-column layout on large screens, two on medium screens, and one on mobile.
This sample system is preloaded in the form. It produces the exact solution x = 2, y = -1, z = 3, and w = 1.
| Equation | x | y | z | w | Constant |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 5 |
| 2 | 2 | -1 | 3 | 1 | 15 |
| 3 | -1 | 4 | 1 | -2 | -5 |
| 4 | 3 | 2 | -2 | 5 | 3 |
Matrix form: A · u = b
Unknown vector: u = [x, y, z, w]ᵀ
Elimination update: Rj = Rj - (ajk / akk) × Rk
This solver applies Gaussian elimination with partial pivoting. Pivoting improves numerical stability by moving the strongest available coefficient into the active pivot position before each elimination stage.
The determinant helps classify the system. A nonzero determinant indicates a unique solution. When the determinant is zero, the rank test distinguishes between no solution and infinitely many solutions.
After triangular reduction, the calculator performs back substitution to recover x, y, z, and w. Residual checks verify how closely the computed values satisfy the original equations.
It solves four linear equations with four unknowns: x, y, z, and w. Each equation must remain linear, so variables cannot be squared, multiplied together, or placed inside nonlinear functions.
A zero determinant means the coefficient matrix is singular. The system may still have infinitely many solutions or no solution. That is why the calculator also compares matrix ranks.
These ranks classify the system. If both ranks are four, the solution is unique. If the augmented rank is larger than the coefficient rank, the system is inconsistent.
Yes. The input fields accept decimal values, negative coefficients, and mixed signs. This makes the page useful for classroom work, engineering models, and numerical experimentation.
A residual measures the difference between each equation’s left side and right side after substitution. Smaller residuals indicate the computed values match the original system more closely.
Partial pivoting improves stability. It reduces rounding problems by choosing the strongest available pivot before eliminating values below it, especially when coefficients differ greatly in size.
Increase precision when your coefficients contain many decimals or when results are very close together. More displayed precision helps reveal subtle differences and smaller residual values.
Yes. After solving, you can download a CSV summary or generate a PDF report. Both options capture the current system status and the main computed metrics.
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.