I have two functions, let's call them $F1$ and $F2$. Both take the same three arguments: $x1, x2$ and $\epsilon$. With every $\epsilon^* \in [0,1]$, there should be either zero or one unique point $(x1^*,\,x2^*)$ that is a solution to
$\qquad F1(x1^*,\,x2^*,\,\epsilon^*) = 0$
$\qquad F2(x1^*,\,x2^*,\,\epsilon^*)= 0.$
What I want to do is make a plot with $\epsilon$ on the x-axis and the two values of $(x1,\,x2)$ on the y-axis so that for every $\epsilon \in [0,1]$ I display the $(x1,x2)$ pair that solves the above equation (if it exists). I know I could do this numerically by gathering points by solving the system for different $\epsilon$ values, but is there a more elegant solution. If not, I would also appreciate help on the numerical solution, since I'm not very comfortable with the Wolfram Language.