The values of $\lambda$ for which the system of equations $x+y-3=0,\left(1+\lambda \right)x+\left(2+\lambda \right) y-8=0,x-\left(1+\lambda \right)y+\left(2+\lambda \right)=0$ is consistent are 

$\Delta =0,{\Delta }_1=0$
$\left|\begin{array}{ccc}1& 1& -3\\ 1+\lambda & 2+\lambda & -8\\ 1& -\left(1+\lambda \right)& 2+\lambda \end{array}\right|=0$
$1\left[{\left(2+\lambda \right)}^2-8\left(1+\lambda \right)\right]-1\left[\left(1+d\right)\left(2+\lambda \right)+8\right]$
$-3\left(-{\left(1+\lambda \right)}^2-\left(2+\lambda \right)\right)=0$
$1\left[4+{\lambda }^2+4\lambda -8-8\lambda \right]-\left[2+\lambda +2\lambda +{\lambda }^2+8\right]$
$-3\left[-1-{\lambda }^2-2\lambda -2-\lambda \right]=0$
$\Rightarrow {\lambda }^2-4\lambda -4-{\lambda }^2-3\lambda -10+3{\lambda }^2+9\lambda +9=0$
$\begin{array}{r}3{\lambda }^2+2\lambda -5=0\\ 3{\lambda }^2+5\lambda -3\lambda -5=0\\ \lambda (3\lambda +5)-1(3\lambda +5)=0\end{array}$
$\lambda =1,\frac{-5}{3}$