I would like to examine whether the following claim is true:
For the following equation: \begin{equation} ax+by=\text{gcd}(a,b) \end{equation}
where $a,b, \in \mathbb{N}$ and $x,y \in \mathbb{Z}$, there exist $x_1,y_1 \in \mathbb{Z}$ such that $x_1 \neq x, y_1 \neq y$ and
\begin{equation} ax_1+by_1=\text{gcd}(a,b) \end{equation} I believe it holds and that it is also possible to express analytically $x_1, y_1$ in terms of $x,y$. I have reached the following point: \begin{align} ax+by=\text{gcd}(a,b) & \Leftrightarrow ax+ax_1-ax_1+by+by_1-by_1=\text{gcd}(a,b) \Leftrightarrow \\ & a(x-x_1)+b(y-y_1)+(ax_1+by_1)=\text{gcd}(a,b) \Leftrightarrow \\ & a(x-x_1)+b(y-y_1)=0 \end{align} where I used the fact that $(ax_1+by_1)=\text{gcd}(a,b)$. After distinguishing cases I end up with $x>x_1, y<y_1$ (or $x<x_1, y>y_1$) and: \begin{equation} x= -(b/a)(y-y_1)+x_1 \end{equation} But from that point on, I cannot figure out how to continue, since when I try to substitute for $y=(\text{gcd}(a,b)-ax)/b$, I end up with just the relation $ax_1+by_1=\text{gcd}(a,b)$.
There must be some wrong step in my proof but I cannot figure it out.