1. Key Concept: Conditions for Consistency of a System of Linear Equations

A system of linear equations in $n$ variables can be represented in matrix form as $AX = B$, where $A$ is the coefficient matrix, $X$ is the column vector of variables, and $B$ is the column vector of constants. For a system of three linear equations in three variables ($x, y, z$), we use Cramer's Rule to determine its consistency.

Let $\Delta = \det(A)$ be the determinant of the coefficient matrix. Let $\Delta_x, \Delta_y, \Delta_z$ be the determinants obtained by replacing the first, second, and third columns of $A$ respectively with the constant vector $B$.

The conditions for the consistency