Unit content
Cramer's rule
For an invertible square system
$$A\mathbf{x}=\mathbf{b}$$
Cramer's rule expresses each coordinate of the solution using determinants. Let $A_i(\mathbf{b})$ be the matrix obtained by replacing column $i$ of $A$ with $\mathbf{b}$. Then
$$x_i=\frac{\det\left(A_i(\mathbf{b})\right)}{\det(A)}$$
The formula has a geometric interpretation: the determinant measures oriented volume, and replacing one transformed basis vector by the target vector isolates the scale factor along that basis direction.
Cramer's rule is especially useful for understanding the connection between determinants and solvability. For larger systems, elimination methods are usually more practical computationally.