Math & Statistics
Linear System Verification Calculator
Evaluate each left-hand side at the entered candidate, compare it with its right-hand constant, combine the residuals, and retain the determinant as a uniqueness warning.
Equation 1 left side-
Equation 2 left side-
Equation 1 residual-
Equation 2 residual-
Largest absolute residual-
System determinant-
Tolerance proximity score-
Residual magnitude reference-
Decision view
Two-equation candidate verification
Two-equation candidate verificationThe candidate is substituted into both equations before signed residual, maximum error, and tolerance score are calculated.
Exact scenario comparisonCandidate x value changes while all other entered assumptions remain constant.
| Candidate x value | Equation 1 left side | Equation 2 left side | Equation 1 residual | Equation 2 residual | Largest absolute residual | System determinant | Tolerance proximity score | Residual magnitude reference |
|---|
How to use Linear System Verification Calculator
- Enter coefficients and constants exactly as written.
- Substitute the candidate into both equations.
- Inspect both residuals and the determinant before accepting the pair.
Calculator guide
Understanding Linear System Verification Calculator
A candidate solution is verified by substitution: both equations must be satisfied, and their signed residuals reveal the direction and size of any mismatch.
Calculation method
How the calculation works
Substitute a candidate x and y into both equations, calculate signed residuals, determinant, maximum error, and a tolerance-proximity score that reaches 100 only at zero residual. Compute a₁x+b₁y and a₂x+b₂y, subtract c₁ and c₂, take the largest absolute residual, and compare it with the entered numerical tolerance.
Verification logic
Distinguish fit from uniqueness
Residuals test the candidate while the determinant describes the equation geometry.
Worked situations
Practical examples
- Residual zero means an equation is satisfied numerically.
- Opposite-signed residuals can cancel in a sum, so each must be checked.
- A zero determinant indicates dependent or inconsistent equations rather than a unique intersection.
Better inputs
Useful tips
- Use a tolerance appropriate to input precision.
- Scale poorly conditioned equations before numerical work.
- Retain more digits than the display when verifying software output.
Before relying on the result
Limitations and common mistakes
- A small residual does not validate the source model or units.
- Near-zero determinants can make solutions sensitive to rounding.
- This page verifies a candidate; it does not solve or condition the system.
Reference
Key terms
- Residual
- Evaluated left side minus right-side constant.
- Tolerance
- Accepted numerical mismatch threshold.
- Determinant
- a₁b₂−a₂b₁ for the two-equation coefficient matrix.
- Conditioning
- Sensitivity of a solution to small input changes.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Does residual zero prove the equations are correct?
No; it only proves the candidate satisfies the entered equations.
Why use maximum absolute residual?
It prevents one badly fitted equation from being hidden by the other.
What does determinant zero mean?
The system has no unique solution.
Does the calculator solve for x and y?
No.