Enter Coefficients and Constants
Use rows as equations and columns as variables x, y, z, w, and u.
Example Data Table
This example system has the solution x = 1, y = 2, z = 1, w = 2, u = 1.
| Equation | x | y | z | w | u | Constant |
|---|---|---|---|---|---|---|
| 1 | 2 | 1 | -1 | 1 | 1 | 6 |
| 2 | 1 | 3 | 2 | -1 | 1 | 8 |
| 3 | 3 | -1 | 4 | 2 | -1 | 8 |
| 4 | 1 | 2 | -1 | 5 | 3 | 17 |
| 5 | 2 | -1 | 1 | 1 | 4 | 7 |
Formula Used
Matrix form: A × X = B
Unknown vector: X = [x, y, z, w, u]T
Unique solution condition: det(A) ≠ 0 and rank(A) = 5
Residual check: r = A × X − B
The solver applies Gaussian elimination with partial pivoting. It transforms the augmented matrix [A|B] into reduced row echelon form, identifies pivots, checks ranks, and extracts the solution when the system has full rank.
If the coefficient matrix is singular, the calculator compares rank(A) and rank([A|B]) to classify the system as inconsistent or dependent.
How to Use This Calculator
- Enter the five coefficients for each of the five equations.
- Type the constant term on the right side of each equation.
- Set the output precision and the pivot tolerance.
- Click Solve Equations to place the result below the header.
- Review the classification, determinant, rank values, and matrix snapshots.
- Use the CSV and PDF buttons to save the computed report.
FAQs
1. What kinds of equations does this solver handle?
It handles five linear equations with five unknowns. Each equation should be written in coefficient form, with constants placed on the right side.
2. What happens when the determinant is zero?
A zero determinant means the coefficient matrix is singular. The system may then have no solution or infinitely many solutions, depending on the ranks.
3. Why does the calculator show rank values?
Ranks help classify the system. Matching ranks below five imply dependence, while different ranks show inconsistency between equations and constants.
4. What is the benefit of residual checks?
Residuals measure how closely the computed solution satisfies each original equation. Values near zero confirm a numerically stable answer.
5. Why is pivot tolerance adjustable?
Tolerance controls when a very small number is treated as zero. This is useful for noisy decimals or nearly singular matrices.
6. Can I enter decimal coefficients?
Yes. The inputs accept decimals, negatives, and integer values. The solver uses floating point arithmetic throughout the elimination process.
7. What do the elimination snapshots show?
They show major row operations after swaps, normalization, and elimination. These snapshots help you audit the matrix reduction path.
8. Is this suitable for symbolic algebra?
No. This page is designed for numeric linear systems. Exact symbolic manipulation would require a different algebra engine.