Implicit function theorem and constrained maximization

Question

For simplicity, consider the following constrained maximization problem:

\begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} & & F(x,y) \\ & \text{subject to} & & x + y = 1, \end{aligned} \end{equation*} with $F(x,y)$ concave. We then have the following system of equations from the first order condition: \begin{equation*} \begin{aligned} & F_1(x,y) - F_2(x,y) = 0\\ & x + y = 1, \end{aligned} \end{equation*} where $F_i(x,y)$ is the derivative wrt the $i$ argument. Now suppose I want to find $\frac{dx}{dy}$ around the solution $(x^*,y^*)$. Using the implicit function theorem however, we get two different expressions depending on wich equation we use:

\begin{equation*} \begin{aligned} & \frac{dx}{dy} = \frac{F_{22}-F_{12}}{F_{11} - F_{21}}\\ & \frac{dx}{dy} = -1, \end{aligned} \end{equation*}

Now my question are:

1. Does it make sense to evaluate the first expression in $(x^*,y^*)$ (thus applying the implicit function theorem and disregarding the second).
2. Why we cannot use the second expression, since we cannot evaluate it at $(x^*,y^*)$?
3. Are there some conditions not satisifed in this setting that prevent us from using the implcit function theorem?