do the parametric equations x=3t^3+7, y=2-t^3, z=5t^3+1 determine a line?
I think they all together determine a line, but I don't know how to explain in a proper way!! please help.

Yes, it's a line. From y = 2-t^3 you have t^3 = 2-y. Sub that back into the other two equations and you have:


x = 3t^3+7 \ = \ 3(2-y)+7\ =\ 13-y \\
z = 5t^3+1\ = \ 5(2-y)+1 = 11-y


From this you can see that x and z are both linear functions of y, and hence z is also a linear function of x. So you have a line.