Key Concept: Conditions for Solutions of a System of Linear Equations

For a system of linear equations in three variables, represented in matrix form as $AX=B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix, we can use Cramer's Rule to determine the nature of its solutions.

Let $\Delta = \det(A)$ be the determinant of the coefficient matrix. Let $\Delta_1, \Delta_2, \Delta_3$ be the determinants obtained by replacing the 1st, 2nd, and 3rd columns of $A$ respectively with the constant terms from matrix $B$.

There are three possibilities for the solutions:

1. Unique Solution: If