If a , b and c are non-zero real numbers and if the system of equations a - 1 x = y + z , b - 1 y = z + x and c - 1 z = x + y have a non-trivial solution, then 3 2 a + 3 2 b + 3 2 c is equal to

Step-by-step solution

System of equations can be written as a - 1 x - y - z = 0 x - b - 1 y + z = 0 x + y - c - 1 z = 0 Using Cramer's rule, for non-trivial solution ∆ = 0 a - 1 - 1 - 1 1 - b - 1 1 1 1 - c - 1 = 0 ⇒ a - 1 b - 1 c - 1 - 1 - - 1 - c - 1 - 1 - 1 1 + b - 1 = 0 ⇒ a - 1 b c - b - c - c - b = 0 ⇒ a b c - a b - a c - b c + b + c - c - b = 0 ⇒ a b + b c + c a = a b c ⇒ 1 a + 1 b + 1 c = 1