Can I use a constraint in solving systems of equations resulting from Lagrangians?

Question

Say that this is a function to minimize: $$F(x,y) = x^4+\frac 1 3 x^3 + \frac 1 2 x^2 - x + y^3$$ subject to the following constraints: $$x^2+x-1=0$$ $$y \geq 5$$ I can introduce Lagrangian multipliers: $$L(x,y,\alpha,\beta) = x^4+\frac 1 3 x^3 + \frac 1 2 x^2 - x + y^3 + \alpha(x^2+x-1)+\beta(y-5)$$ Then, one of the steps is going to be to set $\frac{\partial L}{\partial x} = 0$, i.e.: $$4x^3+x^2+x-1+\alpha(2x+1)=0$$ Since $x^2+x-1=0$ is a constraint, may I replace $x^2+x-1$ in $\frac{\partial L}{\partial x}$ with $0$ to get: $$4x^3+\alpha(2x+1)=0$$ and then proceed with solving the system?

Answer 1

Yes of course we can, indeed the constraint equation is a part of the system to be solved and thus the substitution in the others equations is allowed.