If the system of equations x + y + z = 6,

The system of equations $x+y+z=6, x+2 y+3 z=10, x+2 y+\lambda z=12$ is inconsistent.

$\therefore \Delta=\left|\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda\end{array}\right|=0$

$\Rightarrow 1(2 \lambda-6)-1(\lambda-3)+1(2-2)=0$

$\Rightarrow 2 \lambda-6-\lambda+3=0$

$\Rightarrow \lambda-3=0$

$\Rightarrow \lambda=3$

Also, for $\lambda=3$,

$\Delta_{y}=\left|\begin{array}{ccc}1 & 6 & 1 \\ 1 & 10 & 3 \\ 1 & 12 & 3\end{array}\right| \neq 0$ and $\Delta_{z}=\left|\begin{array}{ccc}1 & 1 & 6 \\ 1 & 2 & 10 \\ 1 & 2 & 12\end{array}\right| \neq 0$

Thus, the value of $\lambda$ is 3 .

If the system of equations $x+y+z=6, x+2 y+3 z=10, x+2 y+\lambda z=12$ is inconsistent then $\lambda=$ 3