Suppose we have two equation
where $c$ is an unknown constant.
I am trying to plot a graph for $x,y$. One way to do it is to solve one of the equations for $c$ then substitute the value of $c$ in the second equation to obtain an expression for $x$ and $y$. Then we can use the ContourPlot to plot a graph for $x$ and $y$.
However, what if $f$, $g$ are too complicated and we can't solve for $c$ to substitute in the other equation to obtain the an expression for $x,y$. Is there another method or a special function one can use to plot without solving the equations?
$f(x,y,c)=c \left(c^3-c x^2+\log (c)+2 (x-1)^2\right)-(c-1)^2 y^2=0$
$g(x,y,c)=2 \left(-c \left(x^2+y^2-1\right)+2 \sin ^3(c)+(x-1)^2+y^2\right)=0$
In Mathematica terms:
f[x_, y_, c_] := c (c^3 - c x^2 + Log[c] + 2 (x - 1)^2) - (c - 1)^2 y^2; g[x_, y_, c_] := 2 (-c (x^2 + y^2 - 1) + 2 Sin[c]^3 + (x - 1)^2 + y^2);