# solution of system of equations

Is it decidable when the following system of equation has solutions over the integers?

$$\begin{matrix} x_1y_2-x_2y_1=a_1,\\ x_1y_3-x_3y_1=a_2,\\ x_2y_3-x_3y_2=a_3. \end{matrix}$$

Here $a_1$, $a_2$ and $a_3$ are integers which $GCD(a_1,a_2,a_3)=1$ and $x_1,x_2,x_3,y_1,y_2,y_3$ are variables.

Using $x_1=0$, $x_2=-1$, $x_3=-1$, $y_1=a_1$, $y_2=a_2$ and $y_3=a_2-a_3$, you have a solution.