Let α β γ = 45 ; α , β , γ ∈ R . If x ( α , 1 , 2 ) + y ( 1 , β , 2 ) + z ( 2 , 3 , γ ) = ( 0 , 0 , 0 ) for some x , y , z ∈ R , x yz  = 0 , then 6 α + 4 β + γ is equal to _______

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​ α β γ = 45 , α β γ ∈ R x ( α , 1 , 2 ) + y ( 1 , β , 2 ) + z ( 2 , 3 , γ ) = ( 0 , 0 , 0 ) x , y , z ∈ R , xyz  = 0 α x + y + 2 z = 0 x + β y + 3 z = 0 2 x + 2 y + γ z = 0 xyz  = 0 ⇒ non-trivial ​ α 1 2 ​ 1 β 2 ​ 2 3 γ ​ ​ = 0 ​ ​ ⇒ α ( β γ − 6 ) − 1 ( γ − 6 ) + 2 ( 2 − 2 β ) = 0 ⇒ α β γ − 6 α − γ + 6 + 4 − 4 β = 0 ⇒ 6 α + 4 β + γ = 55 ​

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