I am afraid there is no "generic" way except for the method you describe yourself: iterate over one variable and solve for the other.
Complications
Note that you have to be careful to find all solutions, not just a solution. This is a major stumbling block in creating a working general algorithm.
Another stumbling block is the singularity points: when f'(x)=0
, you will want to solve for y
and, vice versa, when g'(y)=0
, you will want to solve for x
. What if both are 0 at the same time? You will need to do some paper-and-pencil analysis.
Special Cases
There are some problem-specific simplifications though.
In your specific case the equation for x
is quadratic, so a well known simple closed formula exists. This means that iterating over y
and solving for x
is easier. (The equation for y
is cubic, so a less well known and much more complicated formula exist too).
Another way is to find a parametric representation of your curve (e.g., x^2+y^2=1
is equivalent to x=cos(t); y=sin(t); 0<=t<2*pi
).