1. The problem asks for the necessary conditions for the consistency of a non-homogeneous system of linear equations. 2. A non-homogeneous system can be written as $A\mathbf{x} = \mathbf{b}$, where $A$ is the coefficient matrix, $\mathbf{x}$ is the vector of variables, and $\mathbf{b}$ is the constant vector (non-zero). 3. The system is consistent if there exists at least one solution $\mathbf{x}$ that satisfies the equation. 4. The key condition for consistency is that the rank of the coefficient matrix $A$ must be equal to the rank of the augmented matrix $[A|\mathbf{b}]$. 5. Formally, the system is consistent if and only if: $$\text{rank}(A) = \text{rank}([A|\mathbf{b}])$$ 6. If this condition holds, the system has either a unique solution (if $\text{rank}(A) = \text{number of variables}$) or infinitely many solutions (if $\text{rank}(A) < \text{number of variables}$). 7. If $\text{rank}(A) \neq \text{rank}([A|\mathbf{b}])$, the system is inconsistent and has no solution. 8. In summary, the necessary and sufficient condition for the consistency of a non-homogeneous system is: $$\boxed{\text{rank}(A) = \text{rank}([A|\mathbf{b}])}$$