Let S be the set of all column matrices `[(b_(1)), (b_(2)), (b_(3))]` such that `b_(1), b_(2), b_(3) in R` and the system of equantions (in real variables)
`-x+2y +5z =b_(1)`
`2x-4y +3z =b_(2)`
`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 `[(b_(1)), (b_(2)), (b_(3))] in S?`

Options

A. `x+2y+3z=b_(1),4y+5z=b_(2)andx+2y+6z=b_(3)`
B. `x+y+3z=b_(1),5x+2y+6z=b_(2)and -2x-y+3z=b_(3)`
C. `-x+2y-5z=b_(1),2x-4y+10z=b_(2) and x-2y+5z=b_(3)`
D. `x+2y+5z=b_(1),2x+3z=b_(2)andx+4y-5z=b_(3)`

Answer

A, D