In each part, the solution space of the system is a subspace of R3 and so must be a line through the origin, a plane through the origin, all of R3, or the origin only. For each system, determine which is the case. If the subspace is a plane, find an equation for it, and if it is a line, find parametric equations.



The system is homogeneous (constant terms are all zero) which means that there is a solution (compatible system). There are two cases: a) Rank of coefficients matrix = number of unknowns--->trivial solution x=y=z=0 b) Rank of coefficients matrix < number of unknowns--->indeterminate system(infinite solutions) The coefficients matrix is as follows:⎡⎣⎢148583106−5⎤⎦⎥ As the determinant is -238 we can say the Rank=3 and the solution is the trivial one, x=y=z=0, that is, the origin