# Linear Algebra: Find parametric and vector equations for line y= 3x-1

#### promitheus

##### New member
Hi, just wanted to clarify my working was correct, even though I am getting a different answer from the back of the text book.

Q: Find parametric equations and equation in vector form for the line y= 3x-1

My attempt:

Substituting x=1 and x=0 in the above equation we get two points P(1,2) and Q(0,-1) on the line

Direction vector d= q-p = [-1,-3]

Vector equation is of the form x= p + dt

So x = [1,2] + t [-1,-3] <-----------Question: Could I factor out the -1 as the scalar quantity t covers -1?

Parametric eqn:

x= 1 + t
y= 2 + 3t

The textbook has the co-ordinates of point P and Q switched around.

Am I correct in thinking that the points are arbitrary and should not affect the equations?

Hope this makes sense.
Thanks!

#### tkhunny

##### Moderator
Staff member
Hi, just wanted to clarify my working was correct, even though I am getting a different answer from the back of the text book.

Q: Find parametric equations and equation in vector form for the line y= 3x-1

My attempt:

Substituting x=1 and x=0 in the above equation we get two points P(1,2) and Q(0,-1) on the line

Direction vector d= q-p = [-1,-3]

Vector equation is of the form x= p + dt

So x = [1,2] + t [-1,-3] <-----------Question: Could I factor out the -1 as the scalar quantity t covers -1?

Parametric eqn:

x= 1 + t
y= 2 + 3t

The textbook has the co-ordinates of point P and Q switched around.

Am I correct in thinking that the points are arbitrary and should not affect the equations?

Hope this makes sense.
Thanks!
If all that is required is to trace a line - no other consideration - then it matters not what points you use.

If you need a specific direction or orientation, you'll have to be more careful.

If you are required to be at a specific point at a specific time, you'll have to be more careful.

Can you factor out -1 and let 't' cover it? That makes no sense. When have you ever absorbed a constant with a variable? Anyway, if you do factor it out, and keep it, as should be done, you will change the orientation or the direction or the timing or something.

Of course, whatever your needs, you should not change the sign right at the end after you had been doing so well.

#### promitheus

##### New member
Great thank you. I jumped the gun and changed the sign for the parametric equation.

In regards to the scalar quantity t, I was under the impression that it was a general equation since I had not been given any specific points on the line, and that t was any scalar quantity. But I think I understand why it doesn't work.

Cheers!

#### Jomo

##### Elite Member
Hi, just wanted to clarify my working was correct, even though I am getting a different answer from the back of the text book.

Q: Find parametric equations and equation in vector form for the line y= 3x-1

x= 1 + t
y= 2 + 3t
It is not very difficult to see if you are correct. x= 1 + t => t = (x-1). Substitute into y = 2 +3t => y = 2 + 3(x-1) which does equal = 3x-1 !!

NO, t can not absorb the negative sign! Before you move on you MUST satisfy that for yourself. NEVER accept anything in math! NEVER. You must see it for yourself. Especially if you are in Linear Algebra.