Different types of problems include when the two (or more, but two is complicated as is) variables are each in some operation of the other, such as an exponent, trig.-func., many others, a combination, and recursion (e.g. $x^y=y^x$, $tan(x/y)=sin(y/x)$, $sinx=cosy$, $f(x)=x-f(x(x)$).
Most if not all of these aren’t bijective, but so aren’t many equations without as strong a codependence invertible (such as $x^2-2xy+y^2-8x=-16 ⟺ x=y±2√(2y)+4⟺y=x±√(2x-4)$). What are the cases when an explicit equation definitely is possible, a closed-form less explicit using e.g. product-log function is, certainly neither is, or could be either way depending on finer details?