The system of equations

$$\begin{gathered} -k x+3 y-14 z=25 \\ -15 x+4 y-k z=3 \\ -4 x+y+3 z=4 \end{gathered}$$

is consistent for all k in the set

$$\begin{gathered} \Delta =\left|\begin{array}{ccc}-k& 3& -14\\ -15& 4& -k\\ -4& 1& 3\end{array}\right|=121-k^2 \\ \Delta \ne 0 k\in R-\{11,-11\} (Unique sol.) \end{gathered}$$

If k = 11

${\Delta }_z=\left|\begin{array}{ccc}-11& 3& 25\\ -15& 4& 3\\ -4& 1& 4\end{array}\right|\ne 0$

No solution

If k = -11

${\Delta }_z=\left|\begin{array}{ccc}11& 3& 25\\ -15& 4& 3\\ -4& 1& 4\end{array}\right|\ne 0$

No solution