Let S be the set of all column matrices ⎡⎢⎣b1b2b3⎤⎥⎦such that b1,b2,b3∈R and the system of equations (in real variables)−x+2y+5z=b12x−4y+3z=b2x−2y+2z=b3has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each ⎡⎢⎣b1b2b3⎤⎥⎦∈S?

Let S be the set of all column matrices ⎡⎢⎣b1b2b3⎤⎥⎦such t

1.	Let S be the set of all column matrices ⎡⎢⎣b1b2b3⎤⎥⎦such that b1,b2,b3∈R and the system of equations (in real variables)−x+2y+5z=b12x−4y+3z=b2x−2y+2z=b3has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each ⎡⎢⎣b1b2b3⎤⎥⎦∈S?
Answer» Let $S$ be the set of all column matrices $⎡ ⎢ ⎣ \begin{matrix} b_{1} b_{2} b_{3} \end{matrix} ⎤ ⎥ ⎦$ such that $b_{1}, b_{2}, b_{3} \in R$ and the system of equations (in real variables) $\begin{matrix} - x + 2 y + 5 z = b_{1} 2 x - 4 y + 3 z = b_{2} x - 2 y + 2 z = b_{3} \end{matrix}$ has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $⎡ ⎢ ⎣ \begin{matrix} b_{1} b_{2} b_{3} \end{matrix} ⎤ ⎥ ⎦ \in S$ ?