algebra precalculus - Uniqueness of integers satisfying the Extended Euclidian algorithm

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 $x>x_1, yy_1$) 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.