Systems of linear equations
In matrix form A·x = b. By the Rouché–Capelli theorem the system is consistent when rank A equals the rank of the augmented matrix, and the solution is unique when that rank equals the number of unknowns.
Gaussian elimination
The augmented matrix is reduced by row operations. Gauss–Jordan with partial pivoting is used: the row with the largest absolute pivot is chosen, normalised and eliminated from all other rows. Works for any system, including rectangular and singular ones.
Cramer’s rule
xi = Δi / Δ, where Δ = det A and Δi is the determinant with column i replaced by b. Only for square systems with non-zero determinant. See also the determinant calculator.
Gauss–Seidel
Iterative: xi = (bi − Σj≠i aijxj) / aii with the newest values. Converges for diagonally dominant matrices.
Example
2x₁ + x₂ − x₃ = 8, −3x₁ − x₂ + 2x₃ = −11, −2x₁ + x₂ + 2x₃ = −3 → x₁ = 2, x₂ = 3, x₃ = −1.