Talk:Linear equation

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Normal form[edit]

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[edit]

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'[edit]

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, 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)