Let S be the set of all column matrices  $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$ such that  $b_1,b_2,b_3\in R$ and the system of equations (in real variables)
 $$\begin{gathered} -x+2y+5z=b_1 \\ 2x-4y+3z=b_2 \end{gathered}$$
 $x-2y+2z=b_3$
 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\in S$  ?

$\Delta =\left|\begin{array}{ccc}-1& 2& 5\\ 2& -4& 3\\ 1& -2& 2\end{array}\right|=0$  given equation represent family of lines
$\Rightarrow 2x-4y+3z-b_2+\lambda (-x+2y+5z-b_1)=0$  is same as $x-2y+2z-b_3=0$  for some  $\lambda$
 $\frac{2-\lambda }{1}=\frac{2\lambda -4}{-2}=\frac{5\lambda +3}{2}=\frac{-(b_1\lambda +b_2)}{-b_3}$
  $\lambda =\frac{1}{7}$ and  $13b_3=b_1+7b_2$
 $A\to \left|\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 1& 2& 6\end{array}\right|\ne 0$
 $B\to \left|\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ 2& 1& 3\end{array}\right|=0$ as planes are non parallel is must represent family of planes for solution to exist
$(5\mu +1)x+(2\mu +1)y+(6\mu +3)z=3b_1+\mu 21b_2$ 

Must be 2x + y + 3z – 13 $b_3$= 0 which gives
$\mu =1,b_1+b_2+3b_3=0$  which is true always
  $C\to x-2y+5z=-b_1,x-2y+5z=\frac{b_2}{2},x-2y+5z=b_3$as these are parallel for specific case when  
 $-b_1=\frac{b_2}{2}=b_3$

$D\to \left|\begin{array}{ccc}1& 2& 5\\ 2& 0& 3\\ 1& 4& -5\end{array}\right|\ne 0$