Let $S$ be the set of all column matrices $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right]$ such that $b _ { 1 } , b _ { 2 } , b _ { 3 } \in \mathbb { R }$ and the system of equations (in real variables) $$\begin{aligned}
- x + 2 y + 5 z & = b _ { 1 } \\
2 x - 4 y + 3 z & = b _ { 2 } \\
x - 2 y + 2 z & = b _ { 3 }
\end{aligned}$$ has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right] \in S$ ? (A) $x + 2 y + 3 z = b _ { 1 } , 4 y + 5 z = b _ { 2 }$ and $x + 2 y + 6 z = b _ { 3 }$ (B) $x + y + 3 z = b _ { 1 } , 5 x + 2 y + 6 z = b _ { 2 }$ and $- 2 x - y - 3 z = b _ { 3 }$ (C) $- x + 2 y - 5 z = b _ { 1 } , 2 x - 4 y + 10 z = b _ { 2 }$ and $x - 2 y + 5 z = b _ { 3 }$ (D) $x + 2 y + 5 z = b _ { 1 } , 2 x + 3 z = b _ { 2 }$ and $x + 4 y - 5 z = b _ { 3 }$
Let $S$ be the set of all column matrices $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right]$ such that $b _ { 1 } , b _ { 2 } , b _ { 3 } \in \mathbb { R }$ and the system of equations (in real variables)
$$\begin{aligned}
- x + 2 y + 5 z & = b _ { 1 } \\
2 x - 4 y + 3 z & = b _ { 2 } \\
x - 2 y + 2 z & = b _ { 3 }
\end{aligned}$$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right] \in S$ ?\\
(A) $x + 2 y + 3 z = b _ { 1 } , 4 y + 5 z = b _ { 2 }$ and $x + 2 y + 6 z = b _ { 3 }$\\
(B) $x + y + 3 z = b _ { 1 } , 5 x + 2 y + 6 z = b _ { 2 }$ and $- 2 x - y - 3 z = b _ { 3 }$\\
(C) $- x + 2 y - 5 z = b _ { 1 } , 2 x - 4 y + 10 z = b _ { 2 }$ and $x - 2 y + 5 z = b _ { 3 }$\\
(D) $x + 2 y + 5 z = b _ { 1 } , 2 x + 3 z = b _ { 2 }$ and $x + 4 y - 5 z = b _ { 3 }$