Solve for any of the variables; for instance, [tex]z[/tex]:
[tex]z=1-3x+2y[/tex]
[tex]z=-\dfrac{3-2x-y}3[/tex]
Then
[tex]1-3x+2y=-\dfrac{3-2x-y}3\implies11x-5y=6\implies\begin{cases}y=\frac{11x-6}5\\z=1-3x+\frac{22x-12}5\end{cases}[/tex]
Let [tex]x=t[/tex]; then the intersection is given by the vector-valued function
[tex]\vec r(t)=\left(t,\dfrac{11t-6}5,1-3t+\dfrac{22t-12}5\right)[/tex]
or
[tex]\vec r(t)=\left(t,\dfrac{11t-6}5,\dfrac{7t-7}5\right)[/tex]
