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

  2. $\mathrm{k} \neq 3, \mathrm{~m} \in \mathrm{R}$

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

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


Correct Option: , 4

Solution:

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

$\Rightarrow 24-2(0)-\mathrm{k}(8)=0 \Rightarrow \mathrm{k}=3$

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

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

$=8(4-5 m)$

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

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

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

for inconsistent

$\mathrm{k}=3 \& \mathrm{~m} \neq \frac{4}{5}$

 

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