Why do people write lines ax + by = c instead of y = mx + b?

[flyingpig]

Homework Statement

Yeah that always confuses me, why do books sometiems refer lines implicitly?

 

[olivermsun]

No idea why they do it, but one good reason to write it that way is because there's no "privileged" coordinate in a plane.

Think about the equation for a circle -- it only gets confusing when you try to define y as a "function" of x or vice-versa.

 

[LCKurtz]

ax + by = c is a more general form than y = mx + b. That's probably why it is called the "general form". Not all straight lines can be written in the y = mx + b form. Try to write the vertical line through (5,0) that way.

 

[hunt_mat]

I write the equation of the line $$y=mx+c$$...

It's more general, when you come on to planes which are written $$ax+by+cz=d$$, it's a nice generalisation of the equation of the line.

 

[phinds]

I write the equation of the line $$y=mx+c$$...

It's more general, when you come on to planes which are written $$ax+by+cz=d$$, it's a nice generalisation of the equation of the line.

And how do you deal with the issue that LCKurtz brought up?

 

[hunt_mat]

I was attempting some humour there...

But the point is well said, i normally just write x=k for vertical lines.

 

[Char. Limit]

I was attempting some humour there...

But the point is well said, i normally just write x=k for vertical lines.

Which, of course, can be given from the general form of the line by letting b=0 and then letting k=c/a.

 

[Hurkyl]

The symmetry becomes even stronger if you use projective coordinates -- i.e. if Z is nonzero, then (X:Y:Z) refers to the point (X/Z, Y/Z). In these coordinates, the equation of a line is most conveniently written as:

aX + bY + cZ = 0​

(quick exercise: prove that if (X:Y:Z) and (X':Y':Z') define the same affine point, then either both points satisfy the above equation, or neither does)
(aside: Z=0 corresponds to the "points at infinity" on the "projective plane")