1.

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. The, 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 ?

Answer»

`x+2y+3z=b_(1), 4y+5z=b_(2)` and `x+2y+6z=b_(3)`
`x+y+3z=b_(1), 5x+2y+6z=b_(2)` and `-2x-y-3z=b_(3)`
`x+2y-5z=b_(1), 2x-4y+10z=b_(2)` and `x-2y+5z=b_(3)`
`x+2y+5z=b_(1), 2x+3z=b_(2)` and `x+4y-5z=b_(3)`

SOLUTION :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)`.


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