Key Concept: Conditions for Infinitely Many Solutions in a System of Linear Equations

For a system of linear equations 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 constant terms, we use Cramer's Rule to determine the nature of its solutions.

Let $D = \det(A)$ be the determinant of the coefficient matrix. Let $D_j$ be the determinant obtained by replacing the $j$-th column of $A$ with the constant matrix $B$.

For a system of three linear equations with three variables ($x, y, z$) to have infinitely many solutions, the following conditions must be met simultaneously: