Talk:Linear equation

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 . 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 . 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 version. This is:
 * The vector form $N · (X – X_{0})$ (wrong in version), which is   closely related with matrix form (not said in  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 version). It results from taking a unit normal vector $N$ in the above vector form (not said in  version). It is normally written $x cos &alpha; + y sin &alpha; = c$, where $&alpha;$ is the angle between the line of the solutions and the $x$ axis (wrong in  version).
 * As the comments between parentheses show, 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 $$x\,(y_2-y_1) - y\,(x_2-x_1)= x_1y_2 - x_2y_1$$, 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, redirects here, and this article is also the main article of . 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  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 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)

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.

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)


 * 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)

add template to point to linear function
I previously added this but it was reverted by User:D.Lazard, with the comment, "who can confuse an equation and a function?"

It's me. I confuse linear equation with linear function all the time. The fact that the linear function page talks is essentially an expanded disambiguation page about two things, one-variable linear equations and linear maps, I'm inclined to say that directing users to what is essentially a disambiguation page makes a ton of sense. Maybe phrasing it as would be better. - - mathmitch7 (talk/contribs) 18:02, 7 May 2018 (UTC)


 * I'd like to add that I just found out about WP:HATCHEAP and I feel very vindicated. At the very least, I think the article could include, or perhaps  . - - mathmitch7 (talk/contribs) 22:39, 7 May 2018 (UTC)
 * It is included in "See also" section. No reason for more emphasize for this than for others strongly related subjects. However, it could be added in the body of the article that the solutions of a liner equation are the zeros or the level set of the associated linear function (depending on the definition of linear function that is used). D.Lazard (talk) 07:15, 8 May 2018 (UTC)
 * I understand why you are confused between "equation" and "function". Apparently the author(s) of the article was confused themselves. They was so confused that they need several lines for explaining that an equation that is not a linear equation is non-linear!!! I have edited the lead and the introduction of the section on the case of two variables for clarification. I hope that this will clearer to you. D.Lazard (talk) 15:31, 8 May 2018 (UTC)
 * The new lede is really nice, thanks! I put the  template in the Connection with linear functions section. - - mathmitch7 (talk/contribs) 17:53, 8 May 2018 (UTC)

Differences between General Form and Standard Form of Linear Equations
I have browsed the Internet for several examples of sites that specify what the difference between these two forms of linear equations is. Here is a list of 10 of them, in the hopes that the moderators will not revert my changes to this article again:

https://www.mathsisfun.com/algebra/standard-form.html https://www.mathsisfun.com/algebra/line-equation-general-form.html https://www.mathematics-monster.com/lessons/linear_equations_general_form.html https://www.youtube.com/watch?v=1qCSWcPCH_k http://www.mathwarehouse.com/algebra/linear_equation/standard-form-equation-of-a-line.php https://www.calcunation.com/calculator/general-form-line-equation.php http://courses.wccnet.edu/~palay/precalc/22mt01.htm https://www.khanacademy.org/math/algebra/two-var-linear-equations/standard-form/v/standard-form-for-linear-equations https://www.onlinemathlearning.com/forms-linear-equations.html https://www.physicsforums.com/threads/general-form-vs-standard-form-of-a-line.640797/

If my edits get reverted again, I would like a good explanation why all of these web sites are wrong, especially the Khan Academy one that teaches these forms. — Preceding unsigned comment added by Alcazar84 (talk • contribs) 22:08, 26 August 2018 (UTC)


 * The words "general form" and "standard form" are both very generic; it seems extremely implausible to me that there is in fact a consistent distinction made between these two terms in reliable sources. There are two reasons that your approach here is not promising:
 * you intentionally set out to find sources that made a particular distinction, rather than to determine whether this distinction is in fact made consistently across sources, and
 * none of those links is to a reliable source.
 * The first problem means that your conclusion is subject to confirmation bias and not actually convincing. The second problem means that none of this is usable to support the distinction in WP, anyhow.  --JBL (talk) 00:53, 27 August 2018 (UTC)


 * I respectfully disagree with your reasoning. These two terms are not very generic; they are specific to the types of equations being discussed.  Of all of the web pages I looked at (at least 25 or so), there was only one that showed what I considered to be the wrong definition of the general form.  Every other one correctly defined both terms the way I had stated them.


 * As for you saying that I "intentionally set out to find sources that made a particular distinction," I was not able to find sites that directly compared the two; they just defined one or the other. I will continue to look for sites that directly compare the two, if that is what is needed to have my edits approved.  It was not that I intentionally looked for these sites, this is what I found everywhere.  But finding all of these examples ought to be enough evidence to prove my points, shouldn't it?  As for finding "reliable sources," I am not sure I understand what it is you want me to provide.  I followed the link regarding reliable sources, but it was not clear in describing what I can provide as a reliable source.  I had at least thought that the Khan Academy link would qualify as reliable, since it is an online tool that specifically teaches mathematical concepts.  What should I be looking for?


 * I guess, what I need to know is what specifically I need to find to prove my point to you and show you that the edits I had made are truly correct. I will look for what you need and show it once I find it.  I would appreciate any help you can give me to direct me to finding the right evidence I need to have my edits approved.  -Alcazar84 (talk) 12:20, 28 August 2018 (UTC)
 * An often quoted mathematics textbook would certainy qualify as a reliable source. See if you can find anything with this Google Books search, or with this. At first sight, I don't see anything promising. What I do see, is a phrase "... called standard (or general) form" in this source, and in fact, pretty much all over the place. - DVdm (talk) 12:49, 28 August 2018 (UTC)
 * One of the difficulties here may reside in the divide between mathematics and mathematical education. Mathematically there is no difference between what you wish to call standard versus general forms of the equation. By that I mean that one can convert between these forms using only extremely elementary operations, thus falling below the radar for any mathematician. Math educators, on the other hand, being concerned with introducing the elementary topics, would not be so willing to dismiss these minor manipulations, thus leading to the bifurcation that you have observed. The sites that you have mentioned all seem to come from the education world and uniformity in that realm is not always present. In so far as educators are adopting the terminology of mathematicians, there is an element of chance involved in what they use since the mathematicians consider the terms to be equivalent. --Bill Cherowitzo (talk) 17:45, 28 August 2018 (UTC)
 * Addendum: I should also mention an area where mathematicians do differentiate between "standard" and "general". In the geometry of conic sections, there is a general form for the equation of any conic section and various standard forms for different types of conic sections (ellipses, hyperbolae, etc.). In the general form all possible terms are present, while the standard forms have special "shapes" that permit information to be read off of the equation just by looking at it (like what type of conic section is represented). It is a typical exercise, given a general form, to manipulate it until it becomes a standard form so that specific information can be obtained. If one were to carry this analogy over to linear equations, one of the forms involving the slope of the line should be called the standard form, since you can read information directly from the equation. Unfortunately, this was never done historically and we are stuck with the names that have been handed down.--Bill Cherowitzo  (talk) 21:43, 28 August 2018 (UTC)

Section "Two variables"
I have completely rewritten this section. Here are the issues that I am trying to solve. D.Lazard (talk) 09:57, 3 August 2019 (UTC)
 * The section was written more as an essay or a textbook than as an encyclopedic article. Most of it was devoted to explain very elementary algebraic computations that are far to be specific to the subject of the article. Moreover these computations are not needed for proving the different forms, and simpler proofs can be obtained by direct verification.
 * The structure was confusing: most of the section was devoted to the different forms of the equation of a line, while the study of lines is not the subject of article. Thus I have added a section "Geometric interpretation", and explains the different forms as the forms resulting of different specification of a line.
 * I have removed sections about vectors and matrices, because they involve technical concepts that are not useful here (cross product, area of a parallelogram defined by two vectors). Also, it is confusing to talk of matrices in this case, where only row row and column matrices are involved.
 * I have removed the section about parametric equations: they are not linear equations, at most systems of linear equations. A section on parametric equations of lines must appear somewhere, but certainly not in a section on linear equations of two variables.
 * I have removed the example, which is not about linear equations, but about linear functions.

New Wording Proposal
A linear equation in the variables $$x_1,\ldots,x_n$$ is an equation that can be written in the form:

$$a_1 x_1+a_2 x_2+\ldots+a_n x_n=b$$

where b and the coefficients $$a_1,\ldots,a_n$$

$$a_1,\ldots,a_n$$ are real or complex numbers, usually known in advance. The subscript n may be any positive integer. In most linear algebra courses, n is normally between 2 and 5. In real life problems, n might be 54 or 5400, or even larger. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients $$a_1, \ldots, a_n$$

$$a_1, \ldots, a_n$$ are required to not all be zero.(talk) 03:51, 19 February 2022 (UTC)


 * Which article issue this text is supposed to solve? D.Lazard (talk) 09:31, 19 February 2022 (UTC)
 * A linear equation coefficients that are complex numbers and the article's lead does not have information about this. ScientistBuilder (talk) 14:27, 19 February 2022 (UTC)
 * Looks like it does in the last paragraph. MrOllie (talk) 14:35, 19 February 2022 (UTC)

Additional citations needed?
I came to this article to review the forms of linear equations, not to become involved in an editorial matter. Nevertheless, I'm sufficiently bothered by the six-and-a-half-year-old template message at the top to suggest its removal. Is a consensus for this imaginable? If I have to formulate an argument for this, I would first say that the knowledge contained in the article appears to be of a general and established nature, such that to attribute it to a specific author or authors would seem to be inappropriate, aside from unnecessarily marring the article's appearance and interfering with its legibility. Secondly, how much of the article needs to be sourced? Every paragraph? Every assertion? Finally, if nothing in the article is controversial or contested and it's been thoroughly reviewed over a period of years by persons competent in the subject, I don't see that the addition of a couple of citations would significantly add to its authority. May this disruptive adjuration to add citations be removed, or is it to just sit there perennially inhibiting people from simply reading the article without being previously advised of its defectiveness? –Bret Sterling (talk) 03:11, 14 September 2022 (UTC)

what about methods to solve linear equations
like elimination method, substituting method, square method etc. Yuthfghds (talk) 07:01, 1 June 2023 (UTC)