# Talk:Linear equation

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## FINDING THE SLOPE

THIS SHOULD BE IN THIS PASSAGE PLEASE ADD IT MY SON IS CONFUSED —Preceding unsigned comment added by 216.231.185.94 (talk) 00:30, 14 February 2008 (UTC)

Please do not shout. It is on the page Slope, I believe. Lunakeet 11:59, 12 May 2008 (UTC)

Yeah, dont yell! CAPS equals yell — Preceding unsigned comment added by 75.75.0.38 (talk) 21:59, 9 January 2012 (UTC)

## Side of a line?

Given a point and a line in a plane, how do you determine what side of the line the point is on?

Given an arbitrary plane and a point and line in that plane, how do you even characterize which side of the line is which? It's (relatively) easy if the plane is the usual x-y coordinate system: you can see whether the point is to the left or right, or above or below the line (for example, solve the linear equation for y and plug in the x coordinate of the point, then see if the y coordinate of the point is greater or less than the y given by the equation -- or reverse everything and compare the x values). In higher dimensions (e.g., an arbitrary 2-D plane in 3-D space), it seems that it would be a little harder to even define the problem you're trying to solve. - dcljr (talk) 23:53, 7 November 2005 (UTC)

## Math formatting

For consistency I have changed equations not in-line to Math formatting. Hope there are no problems with this. michaelCurtis talk+ contributions 09:54, 31 May 2006 (UTC)

## Introduction, variables, and formatting

I have a few gripes to express about the current (and aggressively defended) version of the page:

1. The introduction provides an example of a two-variable equation in slope-intercept form. Why? It should provide an example of a generic, multivariable equation, as linear equations are in no way limited to two variables. If it doesn't fall in the introduction (and I think it should), it should at least be covered SOMEWHERE in the article.
2. The article uses atypical variables. For one thing, m, h, and k, as used in this article, are never capitalized (though this article seems insistent upon capitalizing them). For another, h and k are used with conics (representing centers and vertices), not linear equations (which have no centers or vertices). It is common to use (x1,y1) to represent any point on the line (as h and k apply undue significance to that point).
3. Variables should always be italicized. Period.
4. There is no reason to indent the paragraphs of the two-variable linear equation forms.
5. Wording and voice should remain relatively consistent.

I tried to correct most of these, but my edit was quickly reverted. I would like to see some discussion as to why.—Kbolino 16:02, 2 February 2007 (UTC)

Hi Kbolino. Sorry for my perhaps ore hasty revert of your changes. I have some issues with them, but I don't have the time at present to discuss them. So if you feel strongly about them please revert back. Perhaps I will have the time to discuss this later. Regards Paul August 16:48, 2 February 2007 (UTC)

1. I think the introduction of n-dimensional linear equations is better postponed until further along in the article. I have added a new section for this.
2. I think the use of capital letters to designate constants is ok, it is certainly not uncommon. It does help to distinguish the variables from the constants, but I've restored your changes.
3. Yes, I think they all are now.
4. Yes.

I also noticed that when you found the article, it was in a vandalized state, with the slope-intercept form labeled "Point-slope form", and the real slope-intercept form having been deleted, which to your credit you of course fixed.

I hope this addresses all your concerns. If not I am happy to discuss all this further. Again I apologize for my hasty revert.

Regards, Paul August 21:16, 2 February 2007 (UTC)

I appreciate your patience. I can get a bit testy sometimes (that's an understatement, by the way). I understand your reasoning behind capitalizing the constants—in a way, it does make things clearer. I've just never seen them that way and it's foreign to me (which makes them, for me, ironically less clear).
What I meant by the statement about indentation was not a complaint about the headings (I think it was a good idea to have them), but about why they were indented beneath those headings.
As for H and K, I don't fully understand where they came from—I've only ever seen them used with conics. The points I usually see as (x1, y1), (x2, y2)—but this isn't necessarily clear for everybody, and it could lead to confusion with the generic multivariable syntax.
The thing that really got me about capitalization was M (for slope), which I have never in my life seen capitalized (limited range of experience, mind you). I'm not too pragmatic about capitalization—in math, M and m are two different letters to me. It's like throwing g out there for slope.
And the note about voice was in reference to the use of where and here to begin descriptions of formulas. Either one works for me, but the article should try to be consistent; though I'd rather one or the other was used than neither.
I'm going to make an edit to reflect some more of my changes, try to compromise, and see where that turns out. If you don't like it, then please don't just revert—that's what gets me riled up—if one or two things bother you, then just change those things back. I often change things as they scroll by, which means they can easily go unnoticed on reading an edit summary or viewing a diff.—Kbolino 22:55, 2 February 2007 (UTC)
I agree with Kbolino on the matter of presenting the general form of the equation early in the article, with the hyper plane connection. Isn't this what a linear equation defines and is therefore in its essence? In my opinion, the long section on "Forms for 2D linear equations" should not have precedence over the general definition. Regards Knut Vidar Siem 12:02, 24 May 2007 (UTC)

## Normal Form

Is it just me, or does the 'value' to divide by seem more complicated than it really is?

$\frac {-C\sqrt{A^2 + B^2}}{|C|(A^2 + B^2)} = \frac {-C}{|C|\sqrt{A^2 + B^2}}.$

Which really simplifies down to this:

• C < 0 or C = 0
• $\frac {1}{\sqrt{A^2 + B^2}}$
• C > 0
• $\frac {-1}{\sqrt{A^2 + B^2}}$

Actually, now I know the text is technically wrong. It stated to divide by these values when you should actually multiply. Though you could say divide by $\sqrt{A^2 + B^2}$. Also the previous version didn't handle the case where C = 0 since it is in the denominator. I'll fix this article up for now. --Bobcat64 02:07, 7 May 2007 (UTC)

I haven't checked you're point, but here's one thing to keep in mind: it is conventional to "rationalize the denominator"; that is, if you have a radical, put it in the numerator. That's probably why it was the way it was. Dicklyon 02:55, 7 May 2007 (UTC)

## What has differences from linear function

i think (many parts of) this article says about "first-degree polynomial function", and linear function does same one. and is some formulation of this article to be written in the words of linear function (which does not mean a concept of linear maps), isn't it? linear equation can be connected to hyperplane(which contains line, plane, ..., and other codimension-one affine subspaces of Euclidean spaces) or system of linear equations. i suggest "Need clean up" this article and some other related articles. sorry for my poor english, thank you. --218.251.73.163 06:26, 15 May 2007 (UTC) And as well as the fact about finding the slope of a linear should be suggested in this passage.

With regards to the general linear equation at the end of the article, isn't it wrong saying "such an equation will represent a line in n-dimensional Euclidean space"? To my understanding, such an equation represents a hyperplane, not a line; the hyperplane article supports this. Knut Vidar Siem 17:54, 23 May 2007 (UTC)

what the hell is this shit

## Does trigonometry apply to linear equations?

like for example the radius of a circle is equal to

Detailed

radius2 = ( x2 - x1 )2 + ( y2 - y1 )2

This equation represents the length of the radius and it uses the [Pythagorean Theorem]

Just curious --• Storkian • 22:28, 4 October 2007 (UTC)

I don't understand the question; but those equations are not linear, so the answer is probably no. Dicklyon 23:43, 4 October 2007 (UTC)

My bad that was a length equation and it disregarded slope or intercepts. --Storkian aka iSoroush Talk 23:40, 3 December 2007 (UTC)

## Disambiguation?

y=mx+c is taught in the UK why is y=mx+b used here? People who do a search in the UK for example might not be able to find this article. y=mx+b leads here why not y=mx+c?

I did a search for this and found no results except for this article and a few others. Finally finding the linear equation article I realised that this is what I was looking for.

LOTRrules 20:40, 17 October 2007 (UTC)

b and c are just variables that represent the C in the standard equation Ax+By+C=0. maybe b and c should be generalized as C. --• Storkian • 17:13, 2 November 2007 (UTC)
Done. y=mx+c now redirects here. —Celtic Minstrel (talkcontribs) 04:22, 12 December 2007 (UTC)

This reminds me that the TI calculators (TI-83/84) and some text books (Key Math's Discovering Algebra pg 179) list intercept form as y=a+bx, I'd hope my students could figure it out if they visited this site, but I'm not sure. Is it worth mentioning? I don't know. Mrpalmer16 (talk) 21:08, 18 May 2008 (UTC)

As an even more generic way of naming the constants a0 as c, b or whatever, and a1 as m could be used. Oh, and in Sweden kids are taught y=kx+m so I believe that the most generic naming scheme should be used for the sake of clarity. --90.231.217.226 (talk) 17:36, 14 November 2008 (UTC)

## Merge proposal

I suggest merging Linear equation and Linear function. I would prefer to merge to Linear function since it is easy to derive an equation from a function. —Celtic Minstrel (talkcontribs) 20:51, 6 December 2007 (UTC)

Probably better to merge it with System of linear equations. /Pieter Kuiper (talk) 10:04, 9 December 2007 (UTC)
I was a bit too fast there, reacting to the first figure, which shows two linear equations, not a system. /Pieter Kuiper (talk) 10:13, 9 December 2007 (UTC)

what is liniar —Preceding unsigned comment added by 124.104.15.43 (talk) 07:50, 16 June 2008 (UTC)

## Patronising repetition

This article has a vast quantity of cruft. I see that there has been recent vandalism, so I thought to consult first, but there is a huge list of redundant sections saying exactly the same thing about the same equation, with the constants named differently. What on earth is the idea behind that? Also, linear equations are clearly far more general than just lines in R2, though this is not mentioned at all! Without becoming more technical than need be, recall that this is an encyclopaedia, not a school textbook: we inform, not instruct; and can and should have a steep difficultly gradient. Just because we start with trivial statements does not mean that the whole article has to be like that.

Besides, half the names are made up and probably constitute slight original research; they are certainly not mostly full technical terms. Again, in what way is the 'general form' given any more general than the other forms which follow it? (talk) 00:29, 7 February 2009 (UTC)

## clarification of conversions

I don't know how to convert from Ax+By+C=0 form to y=mx+b form. This page didn't help with that. Could someone add a note on the relationship between ABC and mb conventions? —Preceding unsigned comment added by 130.160.63.116 (talk) 00:53, 5 November 2009 (UTC)

I don't know if it belongs on the page, but you just solve Ax + By + C = 0 for y:

Ax + By + C = 0
By = -Ax - C
y = (-A/B)x + (-C/B)

Therefore m = -A/B and b = -C/B. 90.191.161.170 (talk) 08:14, 10 August 2011 (UTC)

## property of linear equation

property of linear equation are to find the unknown variable —Preceding unsigned comment added by 122.52.87.126 (talk) 13:35, 21 January 2010 (UTC)

## HORRIBLE ! Conections w linear functions

This section should be removed if no one can make it right! Here are some undefined, if not just plain incorrect, terms used in that section with NO EXPLANATION: "rules of scalars", "matrix scalar", "accepted slope". The section confuses the graph of the equation with the equation. The section assumes that the domain and range are the same. That is x,y ∈ R. what if y = ix? if x ∈ R then f(y) is undefined. I am not clear what this means: y=f(x) → f(x+y) = f(x) + f(y) -- does this mean that f(x+y) = f(x + f(x)) ?! ANd I have a real issue with the statement that y= ax+b, b≠0 is not a linear function since it doesn't have a zero y intercept. Am I misunderstanding? All in all a botched section. Sorry to be so harsh, but this should be (and could be) a very simple section.69.40.241.198 (talk) 03:38, 3 March 2011 (UTC)

I've rewritten this section to try to address some of the above concerns. Paul August 12:56, 3 March 2011 (UTC)

## Sourcing of forms

Should we work to find sources for each of the forms listed (general, standard, etc...) or are they assumed to be such common knowledge that they're not worth sourcing? Cliff (talk) 23:46, 3 May 2011 (UTC)

## Undid revisions

I reverted several revisions by ArjunMalarmannan (talk). Did not appear intelligible. Qartar (talk) 19:49, 23 July 2011 (UTC)

## Relationship between two equations

Take the following sample: Mary has $87 saved, and earns$3.25 per week. Jane has $89 saved, and earns$3 per week. I know there's a relationship between the equation 87+3.25x and the equation 89+3x, but how do I describe the relationship (besides the fact that both equations equal 113 when x=8)? — Preceding unsigned comment added by 76.181.160.60 (talk) 23:40, 10 February 2012 (UTC)

## Negative slope: inversely related not inversely proportional

Sorry to Paul August that he did not find this following addition by me to be useful here:

"An equation where m is negative, as in $y = -mx + b,\,$ will produce a negative slope. Note that while in this case the two variables are inversely related (as one increases, the other one decreases), they are not "inversely proportional", because one does not halve when the other one doubles.Kirk, T. and Hodgson, N. (2007). IB Diploma Programme: Physics Course Companion. Oxford, UK: Oxford University Press. pp. 20–21. ISBN 978-0199151448."

The reason I thought it was very useful is that: I have not found negative slope to be explicitly mentioned on WP with the point about 'inverse relationship' not being similar to 'inversely proportional'. This is a very common mistake - at least, in non-professional circles. Yes I doubt the point is useful to math professionals or those with strong mathematics education. But linear equations are used by a much larger pool of people, as they are essential for all sorts of work. It is good to come to Wikipedia to discover facts that, while to those 'in-the-know' are eye-rollingly trivial, still need to be pointed out to a wider public. This particular point is not one that can be found easily on the web (try a search). I am happy if the point goes elsewhere, but it should be searchable (and discoverable) somewhere in Wikipedia - preferably somewhere obvious.Stringybark (talk) 13:34, 4 April 2012 (UTC)

A little quibble: when you talk about a negative slope, and you use the equation y=-mx+b, then this m must be positive in order to make the slope parameter (being -m) in that equation negative. You should have said something like "An equation where m is negative, as in y=-3x+b, will produce ..." - DVdm (talk) 15:48, 4 April 2012 (UTC)
You are at liberty to change wording, if you are willing. The real issue here is whether to regard WP as a wide public resource or as a galaxy of club intranets; you make your decision about content on that basis. No further words from me on this page.Stringybark (talk) 23:01, 5 April 2012 (UTC)

## Standard Form??

Where does this nonsense come from? As this form can only be used to describe lines with rational or non-existent slopes, there is nothing standard about it in relation to lines in 2D Euclidean geometry. If working over the rational plane perhaps ... but that is a totally different context. I am going to pull this out unless someone comes up with a reliable reference, and even if that can be done this will need to be modified as it is misleading to say the least. Bill Cherowitzo (talk) 22:50, 12 September 2012 (UTC)

## Whether y = mx+c is linear?

To be linear y = f(x) should satisfy Additivity and Homogeneity Lets check Additivity f(x+y) = f(x) + f(y)

      f(x+y) = mx + my + c
& f(x) + f(y) = mx + my + 2c
So it failed


Lets check Homogeneity f(qx) = qf(x)

      f(qx) = mqx + c
& qf(x) = mqx + qc
Its failed


May be it is nonlinear function Refer https://en.wikipedia.org/wiki/Nonlinear_system

You are confusing between linear function and linear equation. You are right that the function f(x)=mx+c is not linear but affine. But the equation y=mx+c in the two unknowns x and y is a linear equation. D.Lazard (talk) 10:31, 13 April 2013 (UTC)

## A case

The dependence between the length l of a metal bar and its temperature tº is expressed by the formula l= l0(1 + αtº), where l0 is the length of this bar at 0º, and α is a constant.--Barymar (talk) 17:57, 25 April 2013 (UTC)

## A linear equation is NOT a linear function

Sources (other than my own 35 years of teaching mathematics at every level (and a Ph.D. in theoretical mathematics).

A. Constant function is correct.

B1. Linear function is incorrect. This is the definition of a linear mapping from linear algebra on up.

B2. Linear function (calculus) is mostly incorrect; see A. and C. of this note.

D. THIS ARTICLE Linear equation is incredibly misleading. The word linear refers ONLY to the degree of the variables. It has NOTHING to do with line! (This is the implicit implication in this article.)

• Where is a linear equation in 1 variable (unknown), e.g. 3x=6? The solution of set of a linear equation in 1 variable is a value or point on the number line, e.g. x=2. This is the form most children see at ages 7 and they are told that they are solving a linear equation in 1 variable and do not understand why they don't "see" a line (since it is not a linear function and has nothing to do with a line).
• A linear equation in 2 variables and a linear function in 2 variables written in general form are the same, i.e. Ax+By=C. The solution set of both is a set of order pairs (x,y) in the plane that form a line. (I note that it is EXTREMELY unfortunate that teachers in middle school insist that Ax+By=C is a linear "function", since it is very difficult after that for children to understand the idea of a function. In its "starting" form as a "mapping", the definition of a function completely separates the roles of independent and dependent variable, which is obviously not true of a linear function in general form. Unless a child goes on to higher mathematics (and we are talking about less than 10% of the entire population of the entire world), this definition of function, i.e. the explicit form, is the ONLY definition he will ever see. If one does go higher, then one sees functions of all kinds (implicit, parametric, vector-parametric) and can actually categorize linear equations in 2 or more variables as implicit functions that can be solved uniquely in explicit form (and this explanation will definitely help you use mathematics software.)
• Where is the "clarifying" example of a linear equation 3 variables? The solution set of a linear equation in 3 variables is a plane in 3D space. You never mention it, yet every child who completes 12 years of school must solve a system of "3 equations in 3 variables" and has NO idea that he is actually finding the intersection point of 3 planes. This is critical. Again, this is not a linear function and has nothing at all to do with a line (since it is a plane.)

Then you can go on to the general definition. But these three are critical to the definition and understanding of linear equation.

Where is the information that a linear equation is never unique?

Finally, if you properly define linear equation, then you can properly define linear inequality in 1, 2, 3 variables (where OMG you mention linear functional in the second sentence)! (Sources for this: http://www.purplemath.com/modules/ineqlin.htm , http://www.mathplanet.com/education/algebra-1/linear-inequalitites/solving-linear-inequalities , ...)

Lfahlberg (talk) 15:58, 22 June 2013 (UTC)

## Normal form and polar “form”

Whereas the latter has certainly nothing to do with linear equations, I was wrong to refer to both as completely off-topical in [1]. The normal form certainly is a form of a linear equation over real numbers. Though, there are some objections against its presence here in the last revision.

1. It is restricted to real numbers. It has absolutely no sense over rational numbers, for example, whereas all other forms are valid for any field.
2. It uses such concept as trigonometric functions which does not belong to algebra. Also, angle (mod 1 turn, implicitly used) and (arithmetical) square root are not simple things relatively to linear equations.
3. Whereas the problem of parametrizing all lines on the Euclidean plane is obviously meaningful, it is not clear why to parametrize all linear equations on two real variables with one real non-negative parameter and one angular parameter that wraps over 2π (or can take infinitely many values in an arithmetic progression). Which advantages can this form have as an equation? It looks as an unhelpful complication.

If one is willing to explain all this tidily, then feel free to restore the “Normal form” section (but not one about polar coordinates which are inherently non-linear). Incnis Mrsi (talk) 17:39, 28 June 2013 (UTC)

100% agreement from here. - DVdm (talk) 21:03, 28 June 2013 (UTC)

## no citations

At the risk of being again repulsed, i would ask that you look at this page which has 3 citations in english at the beginning. As i have mention several times, this page as it stands incorrectly focus on lines and not on linear equations and that this is a large source of misunderstanding both here and among the readers of such an article. Again, i would ask that the images be added as on the referral page and that Sections on equations of lines to be moved to an article on lines or linear equations. Thank you.Lfahlberg (talk) 06:26, 4 January 2014 (UTC)

Yes, you possibly will be again. Which of three citations supports your
in the case A = 0 ? Note that AxB has the degree 1 if an only if A ≠ 0. Incnis Mrsi (talk) 08:25, 4 January 2014 (UTC)
An interesting observation: Mk.wikipedians actually knew about the A ≠ 0 condition, but Mr. Lfahlberg destroyed it in his first edit to the article. IMHO the situation is evident and further discussion is rather pointless. Incnis Mrsi (talk) 08:35, 4 January 2014 (UTC)
Thank-you Incnis Mrsi for noticing that I had omitted the conditions that the coefficients on the variables be non-zero. This has been corrected. The citations are valid and my other comments are still valid and I would ask that they be considered (and I would ask that responses be posted under the conditions of wikipedia).Lfahlberg (talk) 09:51, 4 January 2014 (UTC)
Clarification: Mr. Lfahlberg refers to this edit.
I wonder: will he ever realize that interleaved Latin letters used in [itex] and their Cyrillic homoglyphs (such as х or А) in running text are strikingly unprofessional? Incnis Mrsi (talk) 10:22, 4 January 2014 (UTC)
I would respectfully request that persons responding here remain on topic. The 3 citations listed on this page give the first level definition of Linear equation and are respectively from the Oxford Concise Dictionary of Mathematics, Mathworld and Purplemath. They are all available online and can be verified in a moment. This definition coincides with the use of the term in the standards in the us common core and the uk curriculum standards. it corresponds to the statement "solve the system of linear equations" that every school system teaches (regardless of whether it is a system of 1, 2, 3 or more linear equations, ...). This article is marked as important. It is not marked as important because a 3rd year student in university theoretical maths needs to check his textbook definition. Lfahlberg (talk) 14:10, 5 January 2014 (UTC)
Sorry, I neglected to look at user:Lfahlberg when the page was eventually created half-year ago. Incnis Mrsi (talk) 07:27, 18 January 2014 (UTC)

In order to give us a broader idea of the term linear equation, I have now compiled a small list of some possible references for citation (given below and with many online links so that they can be easily checked by this audience). This article is important. The article needs to be correct and logically proceed from general mathematics starting with basic algebra, algebra, geometry, linear algebra etc. I would also remark that a there is no such thing in English as a "Line Equation". There are many forms (and formulas) for equations of lines in 2D and 3D (see e.g. Linear function (calculus)).
I truly believe we can make a good and useful article here. I can work on the lower-level parts (up to, but not including linear algebra since I have nearly no memory nor interest in it after 25 years post Ph.D.), but even at the lower level it would be great to work together. I have had really good collaboration within other mathematics articles here on en Wikipedia.
Ok, here is list. Many thanks for reading. Lfahlberg (talk) 16:04, 10 January 2014 (UTC)

At all levels, starting with specific references to general mathematics and working up to generalized linear equations as defined in linear algebra:

Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. pp. 314,315. ISBN 0-8160-5124-0. (Linear equation: An equation is called linear if no variable appearing in the equation is raised to a power different from 1, and no two (or more) variables appearing in the equation are multiplied together. For example, the equation 2x – 3y + z = 6 is linear, but the equations 2x 3 – 5y + z –1 = 0 and 4xy +5xz = 7 are not. A function of one variable is said to be linear if it is of the form f(x) = ax + b, for some constants a and b. More generally, a function of several variables of the form f(x 1 ,x 2 ,…,x n ) = a 0 + a 1 x 1 + a 2 x 2 +…+ a n x n for some constants a 0 , a 1 , a 2 , …, a n is called linear. Any equation of the form ax + by = c represents a LINE in two-dimensional space. (Solving for y, assuming that b is not zero, yields the linear function - ... An equation of the form ax + by + cz= d represents a PLANE in three-dimensional space. A linear combination of variables x 1 , x 2 , x 3 , … is a sum of the form a 1 x 1 + a 2 x 2 + a 3 x 3 +… for some constants a 1 , a 2 , a 3 , … In VECTOR SPACE theory, a set of vectors is said to be linearly dependent if some linear combination of those vectors is zero. In LINEAR ALGEBRA, a MATRIX equation of the form Ax = b is called a linear equation. It represents a system of SIMULTANEOUS LINEAR EQUATIONS....

Carrell, James (2005). "Fundamentals of Linear Algebra". p. 15. Retrieved јануари 2014.

Specific to general mathematics:

Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics" (in англиски). Addison-Wesley. p. 479. Retrieved Септември 2013.

Weisstein, Eric W. "Linear Equation" (in англиски). MathWorld--A Wolfram Web Resource. Retrieved Септември 2013.  (everything in there; albeit not very well explained :))

Specific to one variable:

Marks, Daniel; Cuevas, Gilbert J. (2005). "3 (Solving Linear Equations)". Glencoe Algebra 1, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 69. ISBN 9780078651137.

Dawkins, Paul (207). "College Algebra". Lamar University. p. 63.

Stapel, Elizabeth. ""Solving One-Step Linear Equations"" (in англиски). Purplemath. Retrieved Септември 2013.

Specific to two variables (many references available – I just picked one. I can find more.)

Day, Roger; Hayek, Linda M.; Casey, Ruth M.; Marks, Daniel (2004). "2-2 (Linear Equations)". Glencoe Algebra 2, Student Edition (1 ed.). Glencoe/McGraw-Hill School Pub Co. p. 66. ISBN 9780078656095.

Specific to three variables (I can find actual references):

Specific to linear algebra: (see section below in this talk, et.al.)

Over the complex numbers:

— Preceding unsigned comment added by Lfahlberg (talkcontribs) 16:04, 10 January 2014

## General definition

A linear equation is an equation of the form

$T(x) = b$

where $x \in V$, $b \in W$, $T \colon V \to W$ is a linear map and $V,W$ are vector spaces over the same field (or left/right modules over some unitary ring) [2][3][4]. The article mainly covers the case $V = \R^2, W = \R$, but the concept is, of course, much more general. See Linear Diophantine equation, Linear difference equation, Linear system of equations, Linear differential equation, Linear partial differential equation, Linear integral equation, and so on. Best wishes, --Quartl (talk) 11:28, 4 January 2014 (UTC)

You’re right: currently, the article is a large heap of crap. Obviously, most of authors were able to compile a smooth text with its source given from several textbooks, but it does not demonstrate their ability to think. For instance, look at the “in more than two variables” section. First of all, why more than two (n > 2), not more than one (n > 1)? Is the case n = 2 something special?
Second, why ? Certainly, I know how the number of variables (n) is related to the number of equations (say, m). But is the situation with mn ill-defined? It isn’t; at least no worse defined than Ax = b with a square, but zero-determinant matrix A. Is one linear equation on several variables a pathology? By no means. It is perfectly an equation; it just produces many solutions. On the other hand, must a system of linear equations include necessarily more than one variable? No: for example, all equations in the system may be equivalent to certain linear equation on one variable, but it is still a system, by definition. Incnis Mrsi (talk) 13:19, 4 January 2014 (UTC)
@Quartl: You define what is commonly called a vector equation, or, avoiding too technical terminology, a system of linear equations. The article says explicitly: "This article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients and solutions in any field". Thus it covers (badly, I agree with Incnis Mrsi) the case of a single equation over any field, which is the most common meaning of "linear equation". D.Lazard (talk) 13:55, 4 January 2014 (UTC)
I just wanted address some concerns stated in the sections above by pointing out what the general definition of a linear equation is. Linearity is a well-defined algebraic property which underlies all the cases (and some more) currently covered by the article. The number of variables doesn't really play a role in the definition. Btw., in general the solution of a linear equation is not an element of a field but an element of a vector space.
I know how hard it is to write good mathematics articles (I did write de:Lineare Gleichung by the way). Certainly a good way is to start with the most simple case and then work upwards. Best wishes, --Quartl (talk) 15:40, 4 January 2014 (UTC)
PS: in the German wikipedia we have two articles, one on general linear equations and one on equations for lines (de:Geradengleichung). --Quartl (talk) 15:46, 4 January 2014 (UTC)