Question

If abc = 0, which of the following conditions must a, b
and c satisfy so that the following system of linear
equations has at least one solution?
x + 3y - 4z = a
4x + y - 5z = b
x + y-2z = C
(A) 2a + 3b-5c = 0 (B) 3a + 2b - 5c = 0
(C) 8a + 2b - 11c = 0 (D) 3a + 2b - 11c = 0​

Answer 1

Answer:

....................

Answer 2

Answer:

b-c-a =0

a+b+c =0

b-c+a=0

b+c-a=0

Solution :

We know that, if the system of equations

`a_(1)x + b_(1)y + c_(1)z =d_(1)`

`a_(2)x + b_(2)y + c_(3)z = d_(2)`

`a_(3)x + b_(3)y + c_(3)z = d_(3)`

has more than one solution, then D =0 and `D_(1) = D_(2) = D_(3) =0`. In the given problem,

`D_(1) = 0 rArr |{:(a, 2, 3), (b, -1, 5), (c, -3, 2):}| =0`

`rArr a(-2+15)-2(2b-5c)+3(-3b+c)=0`

`rArr 13a + 4b + 10c -9b +3c =0`

`rArr 13a -13b +13c =0`

`rArr a-b+c = 0 rArr b-a-c =0`

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