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

Answer

A, D

Explanation

Let the given equations represent planes `P_(1), P_(2)` and `P_(3)`.
`Delta=|(-1,2,5),(2,-4,3),(1,-2,2)|=0`
Since no pair of planes is parallel, there are infinite number of solutions.
Let `alphaP_(1)+betaP_(2)=P_(3)`
`:. P_(1)+7P_(2)=13 P_(3)`
`:. b_(1)+7b_(2)=13 b_(3)`
(1) `x+2y+3x=b_(1), 4y+5z=b_(2)` and `x+2y+6z=b_(3)`.
Since `Delta ne 0`, system has at least one solution for any set of values of `b_(1), b_(2)` and `b_(3)`.
(2) `x+y+3z=b_(1), 5x+2y+6z=b_(2)` and `-2x-y-3z=b_(3)`
Since `Delta=0` and `b_(1)+7b_(2) ne 13 b_(3)`, system of equations has no solution.
(3) `-x+2y-5z=b_(1), 2x-4y+10z=b_(2)` and `x-2y-5z=b_(1), 2x-4y+10z=b_(2)` and `x-2y+5z=b_(3)`. Since planes are parallel, there is no solution for any set of values of `b_(1), b_(2)` and `b_(3)`.
(4) `x+2y+5z=b_(1), 2x+3z=b_(2)` and `x+4y-5z=b_(3)`
since `Delta ne 0`, system has at least one solution for any set of values of `b_(1), b_(2)` and `b_(3)`.