The system of linear equations 3x - 2y - kz = 10 2x - 4y - 2z = 6 x...

For inconsistent system of equation the conditions should be $∆ = 0$ and at least one of the ${∆}_{x}or{∆}_{y}or{∆}_{z}$ should not be equal to zero.

Now,

\Delta = \left|\begin{cases}3 & - 2 & - k \\ 2 & - 4 & - 2 \\ 1 & 2 & - 1\end{cases}\right| = 0

$⇒ 24 + 2\left(0\right) - k\left(8\right) = 0 ⇒ k = 3$

And,

{\Delta}_{x} = \left|\begin{cases}10 & - 2 & - 3 \\ 6 & - 4 & - 2 \\ 5m & 2 & - 1\end{cases}\right|

$⇒ {∆}_{x} = \left[10\left(8\right) - 2\left(- 10m + 6\right) - 3\left(12 + 20m\right)\right]$

$⇒ {∆}_{x} = 8\left(4 - 5m\right)$

${∆}_{x} \ne 0 ⇒ m \ne \dfrac{4}{5}$

{\Delta}_{y} = \left|\begin{cases}3 & 10 & - 3 \\ 2 & 6 & - 2 \\ 1 & 5m & - 1\end{cases}\right|

$= 3\left(- 6 + 10m\right) + 10\left(0\right) - 3\left(10m - 6\right)$

$= 0$

{\Delta}_{z} = \left|\begin{cases}3 & - 2 & 10 \\ 2 & - 4 & 6 \\ 1 & 2 & 5m\end{cases}\right|

$⇒ {\Delta}_{z} = 3\left(- 20m - 12\right) + 2\left(10m - 6\right) + 10\left(4 + 4\right)$

$⇒ {\Delta}_{z} = - 60m - 36 + 20m - 12 + 80$

$⇒ {\Delta}_{z} = - 40m + 32$

$⇒ {∆}_{z} = 8\left(4 - 5m\right)$

${∆}_{z} \ne 0 ⇒ m \ne \dfrac{4}{5}$