Chapter 13 - Functions of Several Variables - 13.5 Exercises - Page 913: 21

$$\frac{d y}{d x}=\frac{y-2 x+1}{2 y-x+1}$$

Given $$x^{2}-x y+y^{2}-x+y=0$$ by letting $$F(x,y)=x^{2}-x y+y^{2}-x+y=0$$ So, we have $$F_x(x,y)=\frac{\partial F}{\partial x}=2x-y-1$$ $$F_y(x,y)=\frac{\partial F}{\partial y}=-x+2y+1$$ Also, we get \begin{align} \frac{d y}{d x}&=-\frac{F_{x}(x, y)}{F_{y}(x, y)}\\ &=-\frac{2 x-y-1}{-x+2 y+1}\\ &=\frac{y-2 x+1}{2 y-x+1} \end{align}