This problem asks for the condition under which a given system of linear equations is consistent. This is a classic application of the theory of determinants (Cramer's Rule) or the concept of matrix rank.

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

Consider a system of $n$ linear equations in $n$ variables, represented in matrix form as $AX = B$, where $A$ is the $n \times n$ coefficient matrix, $X$ is the column matrix of variables, and $B$ is the column matrix of constants.

• Case 1: Determinant of the coefficient matrix $D = \text{det}(A) \neq 0$. The system has a unique solution given by Cramer'