The system of linear equations 3 x - 2 y - k z = 10 2 x - 4 y - 2 z = 6 x + 2 y - z = 5   m is…

For inconsistent system of equation the conditions should be $∆=0$ and at least one of the ${∆}_x \mathrm{or} {∆}_y \mathrm{or} {∆}_z$ should not be equal to zero.

Now,

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

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

And,

${\mathrm{\Delta }}_x=\left|\begin{array}{ccc}10& -2& -3\\ 6& -4& -2\\ 5 m& 2& -1\end{array}\right|$

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

$\Rightarrow {∆}_x=8\left(4-5m\right)$

${∆}_x\ne 0\Rightarrow m\ne \frac{4}{5}$

${\Delta }_y=\left|\begin{array}{ccc}3& 10& -3\\ 2& 6& -2\\ 1& 5 m& -1\end{array}\right|$

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

$=0$

${\Delta }_z=\left|\begin{array}{ccc}3& -2& 10\\ 2& -4& 6\\ 1& 2& 5m\end{array}\right|$

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

$\Rightarrow {\Delta }_z=-60m-36+20m-12+80$

$\Rightarrow {\Delta }_z=-40m+32$

$\Rightarrow {∆}_z=8\left(4-5m\right)$

${∆}_z\ne 0\Rightarrow m\ne \frac{4}{5}$