When do parametric equations constitute a line? The given equations specifically are  
$x=3t^3 + 7$
$y=2-t^3$
$z=5t^3 + 3$  
And  
$x=5t^2-1$
$y=2t^2 + 3$
$z=1-t^2$  

There are several ways to see this.  Firstly you can solve each $t^3$ in terms of $x,y$ and $z$, and have a line in terms of its symmetric equation.
You could also re-parameterize these by making the substitution $u = t^3$, and you can see it must be a line in that case.
Incidentally, the second equation is a ray, not a line.  That is because $t^2$ only takes positive values.
