Consider the string 'Let $y = f(x)$." Suppose that it occurs in some elementary context, such as when graphing the function $f$ using $x$/$y$ coordinates. How is this to be understood in predicate logic? We can't have either $x$ or $y$ be free variables, for consider the following:
Let $f:\mathbb{R}\rightarrow \mathbb{R}, \forall x,\ f(x)=2x$
Let $g:\mathbb{R}→\mathbb{R}, \forall x,\ g(x)=x+x$
Let $y = f(x)$
Let $z = g(x)$
$\therefore y = z$
Here the last line is clearly true, but would lack a truth value if either variable were a free variable.
However, if both variables are bound, we're stuck with permutations of quantifiers that mean the wrong things:
$\forall x,\forall y,y=f(x)$ [says the universe has cardinality 1]
$\forall x,\exists y,y=f(x)$ [says f's domain is the universe]
$\exists y,\forall x,y=f(x)$ [says f is a constant function]
$\exists x,\forall y,y=f(x)$ [says the universe has cardinality 1 and f is nonempty]
$\forall y,\exists x,y=f(x)$ [says f is onto the universe]
$\exists x,\exists y,y=f(x)$ [says f is not the empty function]