Least-Squares Solution Calculator
Solve an overdetermined 3-equation, 2-unknown linear system by the normal equations.
Calculator
Enter equations aᵢx + bᵢy ≈ cᵢ. The calculator solves the best-fit x and y.
What this calculator teaches
Least squares finds the best approximate solution when equations cannot all be satisfied exactly.
It is the backbone of regression, fitting, and many engineering estimation problems.
Formula & Symbols
| Concept | Formula | Meaning |
|---|---|---|
| Normal equations | AᵀA x = Aᵀb | Standard least-squares system. |
| Residual sum of squares | SSE = Σ(predicted-observed)² | Measures total squared error. |
Step-by-Step Examples
- The third equation conflicts with the first two.
- Least squares balances all three residuals.
Where students use this
Common Mistakes to Avoid
Check denominator, domain, endpoint, and matrix-invertibility conditions before trusting a result.
Carry extra decimals through intermediate steps and round only the final answer.
Some topics need exact symbolic reasoning, while this calculator gives a clear numerical learning check.
Frequently Asked Questions
Related Math Calculators
Formula Explorer connections
Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.