The system of linear equations
$$\begin{gathered} 3\mathrm{x}-2\mathrm{y}-\mathrm{kz}=10 \\ 2\mathrm{x}-4\mathrm{y}-2\mathrm{z}=6 \\ \mathrm{x}+2\mathrm{y}-\mathrm{z}=5\mathrm{m} \end{gathered}$$
is inconsistent if :

$\mathrm{\Delta }=\left|\begin{array}{ccc}3& -2& -\mathrm{k}\\ 2& -4& -2\\ 1& 2& -1\end{array}\right|=0$
$\Rightarrow 24-2\left(0\right)-\mathrm{k}\left(8\right)=0\Rightarrow \mathrm{k}=3$
${\mathrm{\Delta }}_{\mathrm{x}}=\left|\begin{array}{ccc}10& -2& -3\\ 6& -4& -2\\ 5\mathrm{m}& 2& -1\end{array}\right|$
$$\begin{gathered} =10\left(8\right)-2(-10\mathrm{m}+6)-3(12+20\mathrm{m}) \\ =8(4-5\mathrm{m}) \end{gathered}$$
${\mathrm{\Delta }}_{\mathrm{y}}=\left|\begin{array}{ccc}3& 10& -3\\ 2& 6& -2\\ 1& 5\mathrm{m}& -1\end{array}\right|$
$$\begin{gathered} =3(-20\mathrm{m}-12)-2(6-10\mathrm{m})+10\left(8\right) \\ =40\mathrm{m}-32=8(5\mathrm{m}-4) \end{gathered}$$
For inconsistent
$\mathrm{k}=2\&\mathrm{m}\ne \frac{4}{5}$