Question

The system of linear equations
$3x-2y-kz=10$
$2x-4y-2z=6$
$x+2y-z=5m$
is inconsistent if:

• A. $k=3,m=\frac{4}{5}$
• B. $k\ne 3,m\in R$
• C. $k\ne 3,m\ne \frac{4}{5}$
• D. $k=3,m\ne \frac{4}{5}$

Answer 1

The correct option is D $k=3,m\ne \frac{4}{5}$

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

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

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

$=10\left(8\right)+2(-10m+6)-3(12+20m)$

$=8(7-10m)$
${\Delta }_y=\left|\begin{array}{ccc}3& 10& -3\\ 2& 6& -2\\ 1& 5m& -1\end{array}\right|$

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

$=0$

${\Delta }_z=\left|\begin{array}{ccc}3& -2& 10\\ 2& -4& 6\\ 1& 2& 5m\end{array}\right|$
$=3(-20m-12)+2(6-10m)+10\left(8\right)$

$=40m-32=8(5m-4)$

For inconsistent
$\Delta =0$ and and atleast any one of ${\Delta }_x,{\Delta }_{y}or{\Delta }_z\ne 0$

$k=3$ & $m\ne \frac{4}{5}$