The system of equations k x + y + z = 1 , x + k y + z = k and x + y + z k = k 2 has no solution if k is equal to:

Step-by-step solution

k x + y + z = 1 x + k y + z = k x + y + z k = k 2 Δ = K 1 1 1 K 1 1 1 K = K K 2 - 1 - 1 K - 1 + 1 1 - K = K 3 - K - K + 1 + 1 - K = K 3 - 3   K + 2 = ( K - 1 ) 2   K + 2 For K = 1 Δ = Δ 1 = Δ 2 = Δ 3 = 0 But for K = - 2 , at least one out of Δ 1 , Δ 2 , Δ 3 are not zero Hence for no solution, K = - 2