Calculator
Enter coefficients for equations written in the form ax + by + cz = d. Results appear below the header and above this form after submission.
Example Data Table
This sample system has one unique solution: x = 2, y = 3, z = -1.
| Equation | x Coefficient | y Coefficient | z Coefficient | Right Side | Readable Form |
|---|---|---|---|---|---|
| 1 | 2 | 1 | -1 | 8 | 2x + y - z = 8 |
| 2 | -3 | -1 | 2 | -11 | -3x - y + 2z = -11 |
| 3 | -2 | 1 | 2 | -3 | -2x + y + 2z = -3 |
Formula Used
1. Matrix Form
The solver rewrites the system as A · X = B, where A is the 3 × 3 coefficient matrix, X is the variable vector, and B is the constants vector.
A = [[a₁, b₁, c₁], [a₂, b₂, c₂], [a₃, b₃, c₃]], X = [x, y, z]ᵀ, B = [d₁, d₂, d₃]ᵀ2. Determinant Test
If det(A) ≠ 0, the system has one exact solution.
det(A) = a₁(b₂c₃ - c₂b₃) - b₁(a₂c₃ - c₂a₃) + c₁(a₂b₃ - b₂a₃)3. Cramer’s Rule
For a nonzero determinant, each variable is found by replacing one column of A with B.
x = Dx / det(A), y = Dy / det(A), z = Dz / det(A)4. Rank Check for Singular Cases
When det(A) = 0, the solver compares ranks of the coefficient and augmented matrices.
If rank(A) = rank([A|b]) < 3, infinitely many solutions exist. If rank(A) ≠ rank([A|b]), no solution exists.5. Residual Verification
Residuals measure how closely the computed solution satisfies each equation.
Residual = A · X - BHow to Use This Calculator
- Enter coefficients for the first equation in the form ax + by + cz = d.
- Repeat the same process for the second and third equations.
- Adjust decimal precision, tolerance, graph range, and mesh density if needed.
- Click Solve System to calculate the determinant, ranks, and solution type.
- Review the result section above the form for exact values, matrix reductions, and verification.
- Use the CSV button to save numeric output and the PDF button to download a report.
- Use the 3D graph to inspect how the three planes intersect in space.
FAQs
1. What does this calculator solve?
It solves a system of three linear equations with three unknowns. It also classifies the system as unique, inconsistent, or infinitely solvable using determinant and rank analysis.
2. What happens when the determinant is zero?
A zero determinant means the coefficient matrix is singular. The calculator then checks ranks to decide whether the system has infinitely many solutions or no valid common solution.
3. Why does the calculator show ranks?
Ranks help diagnose singular systems. Matching ranks indicate consistency, while different ranks prove inconsistency. This is the standard test when Cramer’s rule alone is not enough.
4. Can I use decimal coefficients?
Yes. The inputs accept integers and decimals. You can also control display precision and set a tolerance value for near-singular systems affected by rounding.
5. When is Cramer’s rule applied?
Cramer’s rule is used only when the determinant is nonzero. In that case, the system has one exact solution and each variable can be computed directly from Dx, Dy, and Dz.
6. What does the 3D graph represent?
Each equation becomes a plane in three dimensional space. If all planes meet at one point, the system has a unique solution. Otherwise they overlap or miss each other.
7. What can I export?
You can export a CSV file containing the main metrics and solution details. You can also download a PDF snapshot of the result section for reports or notes.
8. Why are residuals useful?
Residuals verify solution quality. Very small residuals indicate the computed values satisfy the original equations closely, which is especially helpful when using decimal coefficients.