Worked example
a: 1 1 2 -1 b: 3 0
→ x = (1; 2)
Method and scope
A may be rectangular; b must be a column with matching rows. Reduction of [A|b] distinguishes a unique solution (∃!), no solution (∅), or infinitely many (∞). Free parameters tᵢ span the displayed nullspace vectors. Residuals alone do not guarantee accuracy for ill-conditioned systems.
rank(A) < rank([A|b]) → ∅; x = x₀ + Σtᵢvᵢ
Precision and limits
≈ marks an approximation. Display rounding does not change the internal calculation.