Question

The value of $\lambda$ for which the system of equations $x+y-2z=0,2x-3y+z=0,x-5y+4z=\lambda$ is consistent is

Answer 1

The given system is:
$x-5y+4z=\lambda$
$x+y-2z=0$
$2x-3y+z=0$

$\Delta =\left|\begin{array}{ccc}1& -5& 4\\ 1& 1& -2\\ 2& -3& 1\end{array}\right|$
$C_1\to C_1+C_2+C_3$
$=\left|\begin{array}{ccc}0& -5& 4\\ 0& 1& -2\\ 0& -3& 1\end{array}\right|=0$

Hence, system is consistant only when${\Delta }_x={\Delta }_y={\Delta }_z=0$.
${\Delta }_x=\left|\begin{array}{ccc}\lambda & -5& 4\\ 0& 1& -2\\ 0& -3& 1\end{array}\right|=-5\lambda =0$
$\Rightarrow \lambda =0$
For $\lambda =0$ clearly ${\Delta }_y={\Delta }_z=0$.
Therefore, system is consistent if $\lambda =0$.