They're really two different types of equal signs.
f(x,y) = x+y might be better written as f(x,y) := x+y where := means "is defined as". Then f(x,y) = 0 is an equation that expands to x+y = 0, or in familiar intro algebra form, y=-x.
I will say that in programming it's commonly used as assignment, which isn't quite the same thing as definition. golang uses it to declare variables so that's pretty close
f(x,y) = x+y might be better written as f(x,y) := x+y where := means "is defined as". Then f(x,y) = 0 is an equation that expands to x+y = 0, or in familiar intro algebra form, y=-x.
g(x,y) := 0 really is a flat plane.