The questions asks for what value of Z does your set of equations have a non trivial solution?
I am not even sure how to begin.. any hints?
A homogeneous system of equations will have a non-trivial solution if and only if the determinant of the coefficient matrix is 0.
Solve $\displaystyle \left|\begin{array}{cccc}-2 & 2 & 5 & 4 \\ -3 & 2 & 5 & 7 \\ 6 & -4 & 5 & 0 \\ 1 & 0 & 15 & Z\end{array}\right|= 0$