# Talk:Linear equation

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## Normal form

The old school class sequence (when I was in high school in 1965) was Algebra, Geometry, Algebra 2, Trigonometry, Analytic Geometry, Pre Calculus.

The Normal form was taught in Analytic Geometry. The underlying form was that a line had direction numbers, and that these could be scaled to be direction cosines. when the direction numbers were scaled to be direction cosines the constant resolved to be the distance to the origin. The absolute shortest derivation of the equations come from Vector Algebra. It is entirely feasible to derive the equation in ordinary college algebra but it is typically long winded. See https://sites.google.com/site/everidgemath/home/writings/algebra document Distance between a point and a line.

The Normal equation of a line has a natural extension to normal to a plane and is typically an early introduction to the concept. The line Ax+By+C = 0 has a normal Vector <A,B>. The particular normal vector from the origin to the line has length C/sqrt(A^2+B^2). The Normal form of the line is a classic which has been taught in analytic geometry for decades.

A particular use of the normal form is this: If N(x,y)=0 is the normal form of the equation of a line then the distance from a point (a,b) to the line is |N(a,b)|. For example the line 3x -4y -5 =0 has the normal form 3/5 x - 4/5 y -1 = 0 and the distance from this line to the point (7,3) is |3/5 * 7 -4/5 *3 -1| =(21-12-5)/5 =4/5

The Normal form is more advanced than most of the other forms in this article, but less advanced than the determinant form or the parametric form. The Normal form for the equation of a line is mentioned (but not completely explicated) in the Wikipedia article Normal Form as The equation of a line: Ax + By = C, with A2 + B2 = 1 and C ≥ 0

The polar form is frequently discussed with the Normal form and is in Wikipedia article Polar coordinate system.

I argue that some bit of the Normal form for the equation of the line should be added to this article. I'd also like to see the polar form as well. This article redirects from equation of a line

This is not new stuff and has been around for ages. Reference Introduction to Analytic Geometry PEECEY R SMITH, PH.D and AKTHUB SULLIVAN GALE, PH.D. GINN BOSTON 1904 pp 92-93 EdEveridge (talk) 20:45, 13 May 2015 (UTC)

Please stop adding this unsourced, poorly written, unencyclopedic, poorly math-formatted content to the article. You added it here before, and it was erased for the same reasons. Second level warning on your user talk page. - DVdm (talk) 21:30, 13 May 2015 (UTC)
I agree with the revert by DVdm. Nevertheless, there is some important material that is lacking in this article and deserve to be added, if it would be better written than in EdEveridge version. This is:
• The vector form N · (XX0) (wrong in EdEveridge version), which is closely related with matrix form (not said in EdEveridge version)
• The normalized standard form or normal form, which, contrarily to the other forms, does not exist for linear equations over other fields than the reals (not said in EdEveridge version). It results from taking a unit normal vector N in the above vector form (not said in EdEveridge version). It is normally written x cos α + y sin α = c, where α is the angle between the line of the solutions and the x axis (wrong in EdEveridge version).
As the comments between parentheses show, EdEveridge version is not the right way for adding this lacking material. D.Lazard (talk) 09:21, 14 May 2015 (UTC)

## Confusing sentence in One Variable section

What does this sentence mean?

"If a = 0, then either the equation does not have any solution, if b ≠ 0 (it is inconsistent), or every number is a solution, if b is also zero."

I'd fix it if I could understand it. The structure is unparseable - If blah, then either blah, if blah, or blah, if blah. Maybe there are too many commas.

I think it should be replaced with a bullet list:

If a = 0, then there are two possibilities:

• b ≠ 0, in which case there is no solution because the equation is inconsistent
• b = 0, where every number is a solution

I have a fondness for bullet lists, though. And this might not even be correct. How can a single equation be inconsistent? Isn't that a term that applies to a system of equations? And how can x = 0/0 possibly be true for all x? Anyway, I'm baffled. — Preceding unsigned comment added by 65.36.43.2 (talk) 16:05, 1 September 2015 (UTC)

I agree, that sentence is awkwardly worded. Your bulleted list is correct, but I think that form is overkill in this situation. I'll rewrite the sentence. Bill Cherowitzo (talk) 16:29, 1 September 2015 (UTC)

## Adding the equation from 'Two-point form' to 'General (or standard) form'

I think it would be useful to add the equation ${\displaystyle x\,(y_{2}-y_{1})-y\,(x_{2}-x_{1})=x_{1}y_{2}-x_{2}y_{1}}$, from the 'Two-point form' section, to the 'General (or standard) form' section. Even though it's stated in the 'Two-point form' section that the equation relates to the General (or standard) form, a reader looking for a way to write the General (or standard) form out of two points would find the equation in the right section right away. GuiARitter (talk) 20:53, 16 December 2015 (UTC)

IMO, it would be even better to remove section "Two point form", and to dispatch its content in the relevant sections General form, Point-slope form and Parametric form (the latter requires also to be completely rewritten, as using notation that is not coherent with that of preceding sections). In fact, there is not really a two-point form, but formulas for getting the various forms from the coordinates of two points of the line.
By the way, Equation of a line redirects here, and this article is also the {{main}} article of Line (geometry) § Cartesian plane. It results that it is very difficult to find the right article, for a reader looking for the equation of a line passing through two points in a space of higher dimension. I suggest to make Equation of a line a true article, and to reduce the corresponding parts of Linear equation and Line (geometry) to a summary with a template {{main}}. D.Lazard (talk) 09:07, 17 December 2015 (UTC)

## "A simple example ... may be expressed as"

In my view this is poor language, so I had undone the edit, upon which user Zedshort (talk · contribs) immediately reverted without any comment. Do we think this is properly expressed? - DVdm (talk) 15:22, 16 August 2016 (UTC)

I was in the process of editing when I submitted and was not trying to revert. Stop being so pedantic and persnikity about the writing of such articles, let's write for people other than mathematicians. People who know the subject will not come here to read such an article. It will be visited by people that are learning the subject and need another source that perhaps expresses the ideas just a little differently. If you come here thinking the purpose is to write for yourself then you have the wrong idea. We should always ask ourselves: "For whom am I writing?" In addition, if a single word of an edit is wrong, then correct that one word, but avoid the wholesale reversion a long string of edits. Doing otherwise suggests that you are squating on the article and are attempting to guard what you believe to be your territory. We human-beasts are very territorial creatures, but we need to overcome such base urges, otherwise there will be endless conflicts both here and in the real world. Zedshort (talk) 15:48, 16 August 2016 (UTC)
I have no comment to this. I'll leave this to the other article contributors. - DVdm (talk) 16:17, 16 August 2016 (UTC)
I'm afraid that that language issue was my fault. I was primarily interested in fixing the formatting of the example and then realized that I should remove the standard form phrase as it was undefined and would have no meaning to a casual reader. As my want, I attempted to edit with the minimum amount of change and that led to the awkward phrasing. With a little more reflection I would have done a better job (and still can, as I see that the formatting needs to be adjusted again.) --Bill Cherowitzo (talk) 17:10, 16 August 2016 (UTC)
Ok, much better, thanks. - DVdm (talk) 19:00, 16 August 2016 (UTC)

## Vector Predicate forms for representing lines (optimized for software geometry)

The section I inserted on "Orientation-Location form" (immediately undone) was an attempt to introduce into this topic a modern, algorithmic geometry (software) perspective. The crux of this new methodology (being taught in Silicon Valley public school) is inventive sketching that results in a sketch specifying an algorithm to be implemented in software.

I plead innocence on the charge of "self-promotion". My goal is to help 21st century math learners pick up the strongest spatial math problem-solving methodologies, which in 2016 implies computational thinking. In the realm of geometry, this means ability to automate your creative solution to a problem by implementing it in software. I understand that this multidisciplinary approach can be unsettling to math teachers who haven't had training in numerical software design and programming. On the other hand, math teachers have an obligation to teach applied Math problem-solving as it is currently being practiced in the real world, and the expectation nowadays is that mathematical thinking be able to be automated (and replicated) via software.

I'm unsure how to proceed how to spread software-savvy Math knowhow using Wikipedia, and welcome suggestions.

The general lack of understanding of a mature spatial computational perspective is becoming an issue in 9-12 Math, where more teachers are bringing Computational Thinking into the classroom, but are stumbling forward unaware of the unique requirements of software math (as compared to math for earlier toolsets, such as paper and pencil + handheld calculator). (BTW, paper and pencil remain essential tools in the computational era). Here are some key changes:

```  • infinity.  Infinity as a numerical value is undefined, and cannot be pushed forward into calculations.   Therefore, in algorithmic math, we seek out representations
that do not depend on infinity as a value.  For instance, the slope-intercept representation of 2D lines is unable to represent vertical lines.
• "="  differentiates into two different concepts, "←" (assignment or information copying) and "==" (predicate evaluation resulting in an equality comparison being true or false)
• chunking information into objects aids in simplifcation, e.g., bundling up the x y z coordinates of a 3D location into a single vector object having its own name.
• representations and algorithms want to be able to handle all cases, with the fewest exceptions (for algorithmic simplicity)
• spatial concepts, representations and algorithms want to be able to scale elegantly going from 2D --> 3D and higher dimensions (if possible)
```
Run direction and orientation of 2D line
2D line equation (predicate form)

The "Orientation-Location form" section I added is similar in its underpinning math pedigree to the "Normal form" (described in this talk page, and also having been controversially deleted from the article). I thought the Wikipedia norm was to err on the side of openness and inclusion (so long as articles don't become redundant). The 1965-era "Normal form" is perhaps a bit outdated for a visual-computational spatial math treatment only in that it doesn't anticipate representing points as vectors, for instance the commonplace by now notation of referring to a 2D point p = [ x y ], and referring to points as p1, p2, p3, etc.

The "normal" discussed gets to one of the (potential) spatial features of a 2D line, the perpendicular vector emanating from the origin out to the line. The only reason this formalism is not perfectly attuned to software computation is that it fails for a tilted line that passes though the origin. The problem with using the "normal" as a feature is that it overcompresses information about line orientation (tilt, slope) with line location in space. In the more modern formulation, the information is split up into orientation and location, and the orientation is stored as a "normalized normal" (unit length direction vector pointing perp. to the line). You can see why a different nomenclature might be advisable, and that's how orientation o has become preferable[1].

Do the mathematicians who view and manage this page want a computational perspective treated in another page?

For example, an article "Line Predicate (computational)"  ??

I can't be the one to decide if more recent, computational refinements to math theory deserve to appear in the Math page, or on a separate page with a reference? But, if that's agreeable, I could take that tack. The main thing is for readers to be able to get to up-to-date Math content, or Computational Math content if you prefer. Pbierre (talk) 20:48, 6 January 2017 (UTC)

References

1. ^ Bierre, Pierre (2010). Flexing the Power of Algorithmic Geometry (1st ed.). Spatial Thoughtware. ISBN 978-0-9827526-0-9.
Wikipedia needs wp:secondary sources. As soon as sufficient scholars find your work sufficiently important, it will be referred to and cited in the relevant literature. Then we can (and probably should) take in on board. A matter of patience. - DVdm (talk) 23:20, 6 January 2017 (UTC)