Lagrange multipliers' question

Let Φ(K, L, λ) = A K^α L^β + λ [P(K) K + P(L) L + M]

∂Φ/∂K = α A K^(α-1) L^β + λ P(K) = 0 --- (i)
∂Φ/∂L = β A K^α L^(β-1) + λ P(L) = 0 --- (ii)
∂Φ/∂λ = P(K) K + P(L) L + M = 0 --- (iii)

rearrange equation (i) and (ii),
α A K^(α-1) L^β = -λ P(K) --- (iv)
β A K^α L^(β-1) = -λ P(L) --- (v)

(iv) / (v) implies
αL / βK = P(K) / P(L) --- (vi)
then P(K) K = α P(L) L / β. Substituting into equation (iii) and solving for L gives this value of L :

L = - β M / [ (α + β) P(L)]

and substituting into equation (vi) and solving for K gives this value of K :

K = - α M / [ (α + β) P(K)]