Consider $(\frac{25}{6},-5,\frac{10}{3})\in\mathbf{Q}^3$.Find all triples $(a_0,a_{1},a_2)$ of relatively prime integers such that

\begin{equation}

(a_0,a_1,a_2)\sim (\frac{25}{6},-5,\frac{10}{3})

\end{equation}

 

Solve:

\begin{align*}

\begin{cases}

a_0=t \frac{25}{6}\\

a_1=-5t\\

a_2=t \frac{10}{3}\\

\end{cases}

\end{align*}

$t\neq 0$.Let $t=\frac{6}{5}k$.

\begin{align*}

\begin{cases}

a_0=5k\\

a_1=-6k\\

a_2=4k\\

\end{cases}

\end{align*}

Let $k=\pm 1$,done.