# The system of linear equations

Question:

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 inconsistent if:

1. (1) $\mathrm{k}=3, \mathrm{~m}=\frac{4}{5}$

2. (2) $k \neq 3, m \in R$

3. (3) $\mathrm{k} \neq 3, \mathrm{~m} \neq \frac{4}{5}$

4. (4) $\mathrm{k}=3, \mathrm{~m} \neq \frac{4}{5}$

Correct Option: , 4

Solution:

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

$3(4+4)+2(-2+2)-\mathrm{k}(4+4)=0$

$\Rightarrow \mathrm{k}=3$

$\Delta_{x}=\left|\begin{array}{ccc}10 & -2 & -3 \\ 6 & -4 & -2 \\ 5 m & 2 & -1\end{array}\right| \neq 0$

$10(4+4)+2(-6+10 m)-3(12+20 m) \neq 0$

$80-12+20 m-36-60 m \neq 0$

$40 m \neq 32 \Rightarrow m \neq \frac{4}{5}$

$\Delta_{y}=\left|\begin{array}{ccc}3 & 10 & -3 \\ 2 & 6 & -2 \\ 1 & 5 m & -1\end{array}\right| \neq 0$

$3(-6+10 m)-10(-2+2)-3(10 m-6) \neq 0$

$-18+30 m-30 m+18 \neq 0 \Rightarrow 0$

$\Delta_{z}=\left|\begin{array}{ccc}3 & -2 & 10 \\ 2 & -4 & 6 \\ 1 & 2 & 5 m\end{array}\right| \neq 0$

$3(-20 m-12)+2(10 m-6)+10(4+4)-40 m+32 \neq 0$

$\Rightarrow m \neq \frac{4}{5}$