Here's a reformulation of the problem in terms of linear algebra. First notice that it is sufficient to solve this problems with coeffficients and x,y,z from the
field . Next, consider two linear transformations
given by the matrices
The problem asks you show that the kernel of B is a subset of the kernel of A. Notice that both linear transformations have rank 1. If the rank-nullity theorem holds for vector spaces over fields with finite characteristic (I am not sure, you'd have to check), then it suffices to find a basis for the kernel of A and check that it lies in the kernel of B.
Well, here's a basis for the kernel of A: (1,1,7) and (0,1,-4). (Assuming rank-nullity, the kernel is 2-dimensional, and these are two linearly independent elements in it, hence a basis.) Both of these vectors lie in the kernel of B, QED.