Are $(\exists x)(\forall y)3x^2y-xy+5y=7$ and $(\forall y)(\exists x)3x^2y-xy+5y=7$ true or false

Question

Consider the statements

$(\exists x)(\forall y)3x^2y-xy+5y=7$

and

$(\forall y)(\exists x)3x^2y-xy+5y=7$

I'm asked to answer if the two statements are true or false for $x,y\in\mathbb{R}$.

For the first statement, I would say false, since for any $x$ we choose, then we'll have a "$..+5y=7$" which won't be true for any $y$ we pick. For the second statement, I would say false, since $y=0$ makes the statement false.

I'm then asked to answer if the two statements are true or false for $x,y\in\mathbb{C}$, however I don't see how that changes anything, by using the same argumentation as above?

Answer 1

All are false. Consider $y=0$.