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સુરેખ સમીકરણ સંહતિ $3 x-2 y-k z=10$ ; $2 x-4 y-2 z=6$ ; $x+2 y-z=5\, m$ સુસંગત ન હોય તો
$k =3, m =\frac{4}{5}$
$k \neq 3, m \in R$
$k \neq 3, m \neq \frac{4}{5}$
$k =3, m \neq \frac{4}{5}$
Solution
$\Delta=\left|\begin{array}{ccc}3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1\end{array}\right|=0$
$\Rightarrow \quad 24-2(0)- k (8)=0 \Rightarrow k =3$
$\Delta_{ x }=\left|\begin{array}{ccc}10 & -2 & -3 \\ 6 & -4 & -2 \\ 5 m & 2 & -1\end{array}\right|$
$=10(8)-2(-10 m+6)-3(12+20 m)$
$=8(4-5 m )$
$\Delta_{y}=\left|\begin{array}{ccc}3 & 10 & -3 \\ 2 & 6 & -2 \\ 1 & 5 m & -1\end{array}\right|$
$=3(-6+10 m)+10(0)-3(10 m-6)$
$=0$
$\Delta_{z}=\left|\begin{array}{ccc}3 & -2 & 10 \\ 2 & -4 & 6 \\ 1 & 2 & 5 m\end{array}\right|$
$=3(-20 m-12)-2(6-10 m)+10(8)$
$=40 m-32=8(5 m-4)$
for inconsistent
$k=3$ and $m \neq \frac{4}{5}$
