Let's consider a linear function of two real variables:
$$ f(x,y) = Ax + By + Cxy + D $$
For any chosen $ x>1$ the function is strictly decreasing in $y$, starting with some positive value and then running down, eventually reaching negative values.
Is there a method to determine/approximate the set of pairs $(x*,y*)$ for which the function f changes sign of its values from positive to negative?
Sure. For any $x \ne -C/B$, set $f = 0$ like this: \begin{align} Ax + By + Cxy + D &= 0 \\ (B + Cx)y &= -Ax - D \\ y &= - \frac{Ax + D}{B + Cx} \end{align} to get $y$.
By the way, for $x = -C/B$, the equation becomes $A(\frac{-C}{B}) + D = 0$, which may be true, in which case the entire line $x = -C/B$, $y = $anything, is part of the zero-set, or it may be false, in which case that line contains no point where the function is zero.