joint optimization problem with somewhat symmetric function

Question

I have just brief question that the method that I use to solve optimization problem is legit. I have function $\max_{x,y}F(x,y)$, and first order condition gives me following equation. $$\frac{\partial F(x,y)}{\partial x}=(x-c')+g(x,y)-z(x,y)=0$$ $$\frac{\partial F(x,y)}{\partial y}=(y-c)-g(x,y)+z(x,y)=0$$ This is just arbitrary functional form.. To solve this equation, what I have done is to multiply $-1$ to the first equation. $$\frac{\partial F(x,y)}{\partial x}=-(x-c')-g(x,y)+z(x,y)=0$$ as both $\frac{\partial F(x,y)}{\partial x}$ and $\frac{\partial F(x,y)}{\partial y}$ are equal to $0$, I set $\frac{\partial F(x,y)}{\partial x}=\frac{\partial F(x,y)}{\partial y}$, then I wil have $$-(x-c')-g(x,y)+z(x,y)=(y-c)-g(x,y)+z(x,y)$$ $$-(x-c')=(y-c)$$ therefore function $F(x,y)$ is maximized at $x=c'$ and $y=c$.