Analyze engineering equations with guided matrix tools. Check solvability, residual errors, and determinant strength instantly. Download reports, compare scenarios, and visualize outputs with confidence.
Each card represents one engineering equation. Enter six coefficients and one constant term.
This sample models a diagonally dominant engineering system. Use the Load Example button to test it instantly.
| Equation | x1 | x2 | x3 | x4 | x5 | x6 | Constant |
|---|---|---|---|---|---|---|---|
| Eq 1 | 10 | 2 | 1 | 0 | 0 | 1 | 33 |
| Eq 2 | 1 | 9 | 2 | 1 | 0 | 0 | 35 |
| Eq 3 | 0 | 2 | 8 | 1 | 1 | 0 | 23 |
| Eq 4 | 1 | 0 | 1 | 7 | 2 | 1 | 47 |
| Eq 5 | 0 | 1 | 0 | 2 | 9 | 1 | 62 |
| Eq 6 | 1 | 1 | 1 | 1 | 1 | 8 | 63 |
A × x = b
Here, A is the 6×6 coefficient matrix, x is the unknown vector [x1, x2, x3, x4, x5, x6]ᵀ, and b is the constants vector.
The solver rearranges rows to place the largest available pivot in the active column. That step improves numerical stability. The matrix is then converted to upper triangular form, and back substitution recovers the six variables.
After solving, the page rebuilds the constants using Ax. The residual for each equation is: Residual = Ax − b. Values near zero indicate a reliable computed solution.
A nonzero determinant usually indicates a unique solution. Rank comparison adds a stronger check. If rank(A) equals rank([A|b]) and both equal 6, the system has one unique answer.
This solver is useful for nodal circuit equations, structural balance models, heat transfer networks, hydraulic flow balance, reaction balance systems, and coupled control models with six unknown quantities.
It solves a linear system containing six equations and six unknowns. The page also reports determinant, matrix rank, reconstructed constants, and residual errors for better engineering verification.
A system can be inconsistent or dependent. Inconsistent systems contain conflicting equations. Dependent systems repeat the same information, so infinitely many solutions may exist instead of one unique vector.
Partial pivoting moves the strongest pivot into place before elimination. This reduces rounding problems and makes the solution process more stable when coefficient values differ greatly in size.
The determinant helps indicate whether the coefficient matrix is singular. A value near zero warns that the system may be unstable, dependent, or too sensitive to small input changes.
Residuals compare the original constants with values rebuilt from the computed solution. Small residuals mean the solved variables satisfy the original equations closely and consistently.
Use a tighter tolerance for clean, well-scaled data. Use a slightly larger tolerance when inputs come from noisy measurements and near-zero pivots should be treated as unstable.
Yes. The inputs accept integers, decimals, and negative values. That makes the page suitable for real engineering models, measured coefficients, and signed balance equations.
The Plotly chart displays the solved values for x1 through x6. It helps you compare magnitude, sign, and relative distribution across the full solution vector.
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.