Three equation lines, one compromise point, and visible residual gapsLive current inputs
Math & Statistics
Linear System Approximation Calculator
Approximate an overdetermined system of three linear equations in two unknowns, quantify every signed residual, and separate best fit from an exact solution.
Least-squares x̂-
Least-squares ŷ-
Sum of squared residuals-
Equation RMSE-
Largest absolute residual-
Normal-matrix determinant-
Fit identifiability-
Exact-versus-approximate status-
CONSTRAINT-SPACE FIT
Three equation lines, one compromise point, and visible residual gaps
Each entered equation becomes a line in x-y space. The highlighted point minimizes the squared algebraic residuals, while short connectors expose which equations disagree most.
RESIDUAL REGISTER
Which equations pull the fitted point away from exact agreement
Signed residuals preserve direction; squared residuals show each equation's contribution to SSE.
| Equation | Entered model | Predicted constant | Signed residual | Squared residual | Share of SSE |
|---|
FIT SETUP
Use comparable equations or normalize them first
- Enter three equations in the same variable order.
- Keep coefficient and constant units compatible across rows.
- Normalize equations if one row is merely scaled much larger.
- Inspect each signed residual rather than only RMSE.
- Treat a near-singular normal matrix as weak identification.
WHAT BEST FIT MEANS HERE
The fitted point minimizes disagreement but does not make the equations true
Three noisy or inconsistent lines usually have no common intersection. Least squares chooses the x and y that minimize the total squared algebraic mismatch.
Scaling an equation changes its leverage. If the equations carry different reliability, a weighted model—not this equal-weight model—is required.
OVERDETERMINED FIT
Solve the normal equations, then audit the residual vector
For A with three rows and two columns, the least-squares estimate solves AᵀAβ = Aᵀc. This minimizes algebraic residuals, not perpendicular geometric distances unless rows are normalized.
Detailed calculation process and general formulas
β̂ = (AᵀA)⁻¹Aᵀcr = Aβ̂ - cSSE = rᵀrRMSE = √(SSE / 3)Aᵀr = 0Symbols, meanings, and units
- A
- 3 by 2 coefficient matrixcoefficient units
- β̂
- fitted vector [x̂, ŷ]ᵀunknown units
- c
- three entered constantsequation units
- r
- signed algebraic residual vectorequation units
- SSE
- sum of squared residualsequation units²
FIT DIAGNOSTICS
Read the solution, error, and leverage together
A small headline RMSE can still hide one dominant row.
Identifiability
-The normal determinant checks whether two independent coefficient directions exist.
Overall disagreement
-SSE and RMSE summarize the residual vector.
Dominant equation
-The register shows the largest individual mismatch.
Decision takeaway: Report the residuals with the fitted pair; an approximation without its error is incomplete.
WHEN TO EXTEND THE MODEL
Signals that equal-weight least squares is not enough
- Different equation variances
- Correlated measurement errors
- Known outlier or bad sensor
- Coefficient uncertainty
- Physical inequality constraints
- Need for robust regression
Applied decisions
Two ways an overdetermined system can mislead
Small balanced noise
All three lines pass near one region and residuals share the burden.
What the result clarifies: The fit point is stable and RMSE summarizes a plausible measurement discrepancy.
One scaled equation
One row is multiplied by a large constant without changing its physical information.
What the result clarifies: Its algebraic residual dominates unless the equations are normalized.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This page performs unweighted algebraic least squares for three equations and two unknowns. It does not estimate measurement uncertainty, choose statistical weights, detect outliers, or validate the underlying physical model.
Linear System Approximation Calculator | Three-Equation Least-Squares Fit FAQ
Why is there usually no exact intersection?
Three measured equations in two unknowns are often inconsistent because of noise, rounding, or model mismatch.
What does Aᵀr = 0 mean?
At the optimum, the residual vector is orthogonal to both coefficient columns.
Can RMSE compare unrelated systems?
Only when equation scaling and units are comparable.
What if the normal determinant is zero?
The coefficient rows do not identify two independent unknown directions, so a unique least-squares pair is unavailable.