Question

The system of equations
$x+y-2z=0$
$2x-3y+z=0$
$x-5y+4z=k$

is consistent for

• A. $k=0$ only
• B. all non-zero real values of $k$
• C. all real values of $k$
• D. two real values of $k$

Answer 1

The correct option is A $k=0$ only

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

$R_2\to R_2-2R_1,R_3\to R_3-R_1$
$\Delta =\left|\begin{array}{ccc}1& 1& -2\\ 0& -5& 5\\ 0& -6& 6\end{array}\right|$

$=5\times 6\left|\begin{array}{ccc}1& 1& -2\\ 0& -1& 1\\ 0& -1& 1\end{array}\right|$

$\Delta =0(R_2\text{and}{R}_3$ are identical)
Hence, system is consistent only when ${\Delta }_1={\Delta }_2={\Delta }_3=0$
Consider ${\Delta }_1=0$
$\Rightarrow \left|\begin{array}{ccc}0& 1& -2\\ 0& -3& 1\\ k& -5& 4\end{array}\right|=0$

$\Rightarrow -5k=0$
$\Rightarrow k=0$
For $k=0$,
clearly, ${\Delta }_2={\Delta }_3=0$
Therefore, system is consistent if $k=0$