Question

Suppose a nonhomogenous system of 9 linear equations and 10 variables has a solution for all possible values. Discuss whether it is possible for two nonzero solutions that are not multiples of each other to exist for the associated homogenous system.

Answer 1

As system has solution for all possible value, then it has solution for zeros in
right part, i.e. for homogenous system.
Solutions of associated system will
be dependent from one fixed variable. Thus, value of this variable
determine
the whole solution. So, all solutions will be linearly dependent,
and answer on your question is negative.