From a computational point of view, Gaussian elimination is generally more efficient than Cramer’s rule for solving systems of equations with more than two or three unknowns. Cramer’s rule requires the computation of n+1 determinants for a system of n equations in n unknowns, while Gaussian elimination has the same computational complexity as the computation of a single determinant. The time complexity of Cramer’s rule is O(n! × n) with a naive determinant-finding algorithm and O(n^4) with LU decomposition, while Gaussian elimination has cubic complexity O(n^3).

Here is an example for both methods:

Consider the system of equations: x + y = 3 2x - y = 2

Using Cramer’s rule: D = | 1 1 | | 2 -1 | = -1 - 2 = -3 Dx = | 3 1 | | 2 -1 | = -3 - 2 = -5 Dy = | 1 3 | | 2 2 | = 2 - 6 = -4 x = Dx/D = (-5)/(-3) = 5/3 y = Dy/D = (-4)/(-3) = 4/3

Using Gaussian elimination: Augmented matrix: [1 1 | 3] [2 -1 | 2] R2 - 2R1 -> R2: [1 1 | 3] [0 -3 | -4] R2/(-3) -> R2: [1 1 | 3] [0 1 | 4/3] R1 - R2 -> R1: [1 0 | 5/3] [0 1 | 4/3] Solution: x = 5/3, y = 4/3

Both methods give the same solution, but Gaussian elimination is generally more efficient for larger systems of equations.
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