General form vs Standard form of a line

[lordofpi]

I'm in the middle of a calculus course (this is not a calculus question per se), studying from the Larson text, and when an answer to a problem is the equation of a line, I solve for Standard form of a line $Ax + By = C$.

I noticed, however, that the Larson textbook frequently terms answers of the equation of a line in the form $Ax + By + C = 0$, which apparently is the General form (based on my searching around on line. This answer is frequently the equation of a line tangent to a given point on a curve, if that is of any interest.

What are the advantages of one form over another? Should I be using General when doing calculus for some reason? Thanks!

 

[Mark44]

I don't see much of an advantage for either form over the other. The first form is slightly more useful for graphing, although the slope-intercept form is probably better yet for nonvertical lines.

The second (general) form has a counterpart with terms up through the second degree, for the conic sections, so maybe that's a motivation for this form.

Overall, the distinctions between standard form and general form here aren't very important, IMO.

 

[Studiot]

What form is the most appropriate to use depends upon what you want to do with this equation.

if you have two expressions of the type f(x,y) = 0 and g(x,y) = 0 you can equate them directly.

If f(x,y) = a and g(x,y) = b then you cannot do this so easily.

For calculus (and many other purposes) I would think that the intercept form of the line is more useful viz

y = mx + b

Since calculus is about slopes and m is the slope.

go well

 

[lordofpi]

Thank you both; that sort of makes sense. I always made use of the slope-intercept form ($y=mx+b$ fairly consistently as well. It just seems to be the most useful of all of the forms. Also, I was very lazy in grade school (decades ago), so -- while I have always had a strong aptitude and love of math -- there are some very interesting gaps in the some of the finer details of things I technically should know lol (like I am actually even relearning how to work with Standard form).

I know Larson is far from the consummate text on calculus, but I figured the authors must have had some motivation for drafting answers in the way that they did. Thanks.

 

[chiro]

Hey lordofpi and welcome to the forums.

The second form is typically the form of the equation of an n-dimensional linear object (also an n-dimensional plane) and it has the same form of n . (r - r0) = 0 for an n-dimensional vectors n, r, and r0 (all have to be the same size, but that size is variable).

In a linear context, this can be useful depending on what you are trying to do.

 

[lordofpi]

Thanks chiro for the additional info. And thank you: this place seems to pickup where Usenet left off all those years ago (plus Usenet never had $\LaTeX$!). I am very excited to be able to participate in the ongoing conversations of so many sharp minds.