to determine the solution of the system:

calculate the ranks of matrix

M1 = A1 B1 C1  D1; ...

and

M2 = A1 B1 C1; A2 ...

The solution exists if and only if rank of M1 = rank of M2. The linear dependence in M1 = two plans are the same. Linear dependence in M2 but not M1 = two plans parallel ...