Edit: Nvm this post. I figured out what was wrong. doh. Need to figure out how to delete this....

We have two lines. l1 goes trough the point (3,1) and is parallel with the vector (2,5). Its equation therefore is:

x = 3 + 2t

y = 1 + 5t

The second line l2 goes through the points P1 = (2,1) and P2 = (1, 2). To be able to write its equation we need a vector which is parallel with the line, and P1P2 = (-1,1) is such a vector. Its equation thefefore is:

x = 1 - t

y = 2 + t

But here is my question regarding the equation of the second line. How to know which point P1 or P2 to use to create the equation? Any point that is on the line l2 is supposed to satisfy the equation of the line, and if that is true, should we not be able to pick P1 instead of P2 to create the equation? That is, both the system:

x = 1 - t

y = 2 + t

as well as the system

x = 2 - t

y = 1 + t

should be valid equations for the line?

If we must be careful about which point we pick, why? How to know which point to pick? Feel like I am missing something obvious here.