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Cramer's rule solves a square system by determinants: replace the column belonging to a variable with the constants, take the determinant and divide by the coefficient determinant.
Cramer's rule
xₖ = det(Aₖ) / det(A), where Aₖ has column k replaced by the constants
D = −2, and the three substituted determinants give x = 1, y = 2, z = 3.
No. It needs n + 1 determinants, so Gaussian elimination is vastly faster beyond about 3 × 3.