Equation of Line: 3x + 2y + 1 = 0 | Coordinates of Third Point On Line

[lolimcool]

Homework Statement

assume the existence of a fixed coordinatization of the plane in which 0 is the origin. Find an equation and give the coordinates of a third point that is on the line a:(-3,4) b:(1,1)

Homework Equations

The Attempt at a Solution

ok so i get x = -3 + t(-1 3)
y = 4 + t(1 -4)
then i solve for t and make the two equation equal each other
and i get 3x +2y + 1 = 0
is that right?


and could someone explain what the existence of a fixed coordinatization of the plane is?

thanks

 

[micromass]

Hi lolimcool!

ok so i get x = -3 + t(-1 3)
y = 4 + t(1 -4)

What do you mean with (-1 3)?

then i solve for t and make the two equation equal each other
and i get 3x +2y + 1 = 0
is that right?

I'm sorry, that equation is not correct If I substitute (1,1) in there, I get 3.1+2+1+1=6 and this had to be 0. (-3,4) does lie on the line, though...

and could someone explain what the existence of a fixed coordinatization of the plane is?

I guess it's just a fancy way of saying that every point of the plane has a set of coordinates. So for example, every point can be given a coordinate (3,4). They make it sound harder than it is...
thanks[/QUOTE]

 

[lolimcool]

Hi lolimcool!


What do you mean with (-1 3)?



I'm sorry, that equation is not correct If I substitute (1,1) in there, I get 3.1+2+1+1=6 and this had to be 0. (-3,4) does lie on the line, though...



I guess it's just a fancy way of saying that every point of the plane has a set of coordinates. So for example, every point can be given a coordinate (3,4). They make it sound harder than it is...
thanks

[/QUOTE]

sorry i mean x = -3 +t(-1,3)
y = 4 + t(1, -4)
which gives me x = -3 + 2t and y = 4 + -3t
i then isolate for t, make the equations equal to each other and i get 3x + 2y + 1 = 0

 

[micromass]

sorry i mean x = -3 +t(-1,3)

But that makes no sense. How would you define -3+(-1,3)?? You can't add up a real number and a pair!

What did you really mean?

 

[lolimcool]

But that makes no sense. How would you define -3+(-1,3)?? You can't add up a real number and a pair!

What did you really mean?

oh i don't know then, i was just kinda following my notes(i obv copied them down wrong)
how would i go about doing the first step then?

 

[micromass]

The equation of a line through (a,b) and (c,d) is given by

$$\left\{\begin{array}{c} x=a+t(c-a)\\ y=b+t(d-b)\\ \end{array}\right.$$

So this is the formula you will have to use...

 

[Mark44]

Given two points on a line,
1) find the slope of the line.
2) find the equation of the line by using the point-slope equation.

For example, if your points are (1, 1) and (2, 3), the slope of the segment joining these points is m = (3 - 1)/(2 - 1) = 2

Using the slope and the point (1, 1), we have
y - 1 = 2(x - 1)
==> y = 2x - 2 + 1
==> y = 2x - 1

 

[lolimcool]

The equation of a line through (a,b) and (c,d) is given by

$$\left\{\begin{array}{c} x=a+t(c-a)\\ y=b+t(d-b)\\ \end{array}\right.$$

So this is the formula you will have to use...

OH K, so that's what i initially did then
i had x = -3 +t(3-1) => x = -3 +2t
y = 4 + t(1-4) => y = 4 -3t

then i got
t= (x+3)/2 and t = (y - 4)/ -3
making both equations equal each other i get
-3x -9 = 2y -8
-3x -2y -1 = 0
3x +2y + 1 = 0

 

[micromass]

OH K, so that's what i initially did then
i had x = -3 +t(3-1) => x = -3 +2t
y = 4 + t(1-4) => y = 4 -3t

No, because I don't see how you found (3-1)...
Your c=1 and a=-3 here...

 

[lolimcool]

No, because I don't see how you found (3-1)...
Your c=1 and a=-3 here...

wow i screwd up
so x = -3 + 4t
and y = 4 - 3t

solved it and got -3x -4y + 7 = 0
which i believe is right
thanks for your help :P

 

[micromass]

Yes, that's good! Well done!

 

[Mark44]

wow i screwd up
so x = -3 + 4t
and y = 4 - 3t

This seems like the long way around to me, writing parametric equations for the line, and then eliminating the parameter to write the equation in standard form. It's not wrong, just extra work that I don't believe is needed.

The suggestion I gave in post #7 is a lot more direct.

solved it and got -3x -4y + 7 = 0
which i believe is right
thanks for your help :P