Bezout's identity: which half is larger?

Question

Bezout's identity: Let a and b be integers with greatest common divisor d. Then, there exist integers x and y such that ax + by = d.

Is it true that if a > b then ax < by? Is there a proof for this?

Answer 1

$x,y$ are not uniquely defined. Let $a=2,b=1$. Then we could take $x=1, y=-1$, which makes your inequality false. Or we could take $x=-1.y=3$ which would make it true.

Note: In the comments, the OP has made clear that $x,y$ are intended to be the "minimal" solution, that is, the solution generated by the Euclidean Algorithm. In my example, the minimal solution is, of course, $x=0, y=1$ for which the claim holds. But even with this extra requirement, however, the general claim is false. For $a=3,b=2$, the minimal solution is $x=1,y=-1$, say.