General form vs Standard form of a line

AI Thread Summary
The discussion focuses on the differences between the Standard form (Ax + By = C) and General form (Ax + By + C = 0) of a line, particularly in the context of calculus. While the Standard form is noted for its utility in graphing, the General form is recognized for its application in higher-dimensional contexts and conic sections. The slope-intercept form (y = mx + b) is highlighted as the most practical for calculus, emphasizing the importance of slopes. Participants agree that the choice of form depends on the specific mathematical needs, with no significant advantages of one over the other for most purposes. Ultimately, understanding these forms enhances mathematical proficiency and problem-solving skills.
lordofpi
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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!
 
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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.
 
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
 
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.
 
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.
 
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.
 
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