Elementary Methods in Number Theory Exercise 1.2.3

Find integers $x$ and $y$ such that
\begin{equation}
13x+15y=1
\end{equation}

Solve:
\begin{equation}
15=1\times 13+2
\end{equation}
\begin{equation}
13=6\times 2+1
\end{equation}
So
\begin{equation}
13=6\times (15-13)+1
\end{equation}
So
\begin{equation}
7\times 13- 6\times 15=1
\end{equation}