Newton-Raphson Method with Sum Constraint

Question

I have a system of non-linear equations $\pmb{f}\left(\pmb{x}\right)$ for which I want to find the approximate root $\pmb{x}^*$ using the Newton-Raphson method, $$\pmb{f}\left(\pmb{x}^*\right) \approx \pmb{0}.$$ I also require the following condition to be satisfied $$\left(\sum_ix_i^*\right)-k = 0$$ for a user-set $k$. I have added this constraint as another row to the system of equations, and updated the Jacobian accordingly. The approximate root I yield, however, does not satisfy the sum constraint. I cannot determine why this is occurring, and would greatly appreciate suggestions for other approaches to solving the root-finding problem with the sum condition.