The non-homogenous system is as follows: $$3x+2y+5z=10\\ 3x-2y=7\\ 6x+4y-10z=k$$
I have determined that: $$z=1-\frac{k}{20}\\ y=\frac{k}{16}-\frac{1}{2}\\ x=2+\frac{k}{24}$$
What are the values of $k$ to form a matrix which is inconsistent and then consistent?
Let $K$ be a field and $k\in K$. Then we can write the system of linear equations in matrix form $Av=b$ with $$ A=\begin{pmatrix} 3 & 2 & 5 \\ 3 & -2 & 0 \\ 6 & 4 & -10 \end{pmatrix},\quad v=\begin{pmatrix} x \\ y \\ z \end{pmatrix},\quad b=\begin{pmatrix} 10 \\ 7 \\ k \end{pmatrix} $$ The system has a unique solution iff $\det(A)\neq 0$ in $K$. In this case the solution is $v=A^{-1}b$. Since $\det(A)=120$ is nonzero for all fields of characteristic not equal to $2,3,5$, we obtain a unique solution $$ v=A^{-1}b=\begin{pmatrix} \frac{48+k}{24} \\ \frac{k-8}{16} \\ \frac{20-k}{20} \end{pmatrix} $$ for any given $k\in K$. The system may be inconsistent for characteristic $2,3,5$. For example, the last equation for $k=1$ and $char(K)=2$ is inconsistent: $0=1$.