Talk:Simultaneous equations

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General structure of article[edit]

I came to this article because I was willing to write an article (which is lacking) on polynomial systems of equations. For this it was natural to start by reference to articles defining the basic facts like "What is an equation?", "What means solving an equation or a system of equations?". I have been unable to find any article dealing to these questions, the present one being the one which is the closest from what I need.

The article is implicitly restricted to a particular set of equations, namely those whose unknown values have to be found in a field. This excludes many important classes of equations and systems of equations: boolean equations, functional equations, differential and integral equations, Diophantine equations,...

My suggestion, which I am unable to realize myself because of lack of time and of experience of Wikipedia:

  • To write an article entitled "Equations and systems of equations" which answers to above questions and present the main classes of equations and systems, with links to articles devoted to them.
  • To write articles on
    • Linear systems. The present article could be a starting point, as the methods which are presented here (substitution and elimination) work generally only in this case. By the way, they are both essentially equivalent to Gaussian elimination.
    • Multivariate polynomial systems: I am willing to write it, at least the part devoted to algebraic methods.
    • Transcendental equations and systems
    • and many other

D.Lazard (talk) 16:50, 19 May 2010 (UTC)

Material moved out of another page[edit]

I have removed the following from the page Substitution method, which deals with an optical technique. I was going to create a new page for this, but perhaps it belongs here or in some other existing article. I'll leave it up to the math folks to figure out where this best belongs--Srleffler 03:51, 11 March 2006 (UTC):

The substitution method is an algebraic method for solving a system of equations (finding the point where two graphed lines intersect). The substitution method, unlike the elimination method, will solve for any type of system, whereas the elimination method will only solve for linear systems. In the substitution method, you do not have to have the equations in the same form. The substitution method infers that since the same variables are used, they equal the same thing. For example, for functions, two y's in an equation, although they may differ in the terms that they are said to be dependent upon, must equal the same thing. Take the equations y = x and y = 2x - 10. Since the same variables are used, they can be easily substituted. In this set of circumstances, you should take one equation (y = x for here), and substitute it in for the y in the other equation. You receive: x = 2x - 10. Then, simply solve it for x. Since there is a difference of 10 between the two, you should receive x = 10. Then, take the other equation (y = 2x - 10), and substitute 10 in for the x's. You should receive: y = 2(10) - 10. This simplifies to 20 -10 = y. Therefore, y = 10. The solution for the system is the ordered pair (10, 10). Always remember to substitute into DIFFERENT equations when you have solved for the first variable. You will not receive a correct answer if you substitute into the same equation.


Expert needed[edit]

If you're an expert you'll see for yourself this article has "much room for improvement".

  1. It is fine that it starts with an example, but then it should also give the solution for the example in the introductory paragraph, where perhaps something more elementary than the present example is better -- such as (x+2y = 7 & xy = 6) with solutions (x = 3 & y = 2) and (x = 4 & y = 1.5).
  2. The intro should not immediately talk about a geometric interpretation. If this is done (later), it should be explained what the relation is between these equations and geometry. This is mostly useful only when there are two or perhaps three real-valued unknowns, but generally not for four unknowns or two complex unknowns.
  3. There should be some clarification of the domain over which the unknowns can range. Is this a system of Diophantine equations? Are we constrained to real numbers or can we have complex solutions? Some other domain? We could also have a system of differential equations, in which the unknowns are functions.
  4. The discussion between the # of equations and the # of variables is incomplete and partly wrong. What does it mean that "every variable will have an explicit solution set"? Aren't we solving simultaneously? What is the meaning of "explicit" here? Why isn't this the case when y>x (except for the finiteness)? Examples are needed here. There should be some discussion of the possibility that there is no solution at all. The claim as stated is "somewhat truish" for the real & complex domains, but not at all for integers.
  5. It should be made clear that there exists no general method for solving equations -- let alone simultaneous equations.
  6. It is not true that systems of simultaneous linear equations "can always be solved" if that means: we can always find solutions. This false interpretation may be the "obvious" interpretation of this sentence for most innocent readers.
  7. On the other hand, elimination is sometimes possible for non-linear equations. If x = r cos φ and y = r sin φ, then x2 + y2 = r2, eliminating φ. Likewise, x/y = tan φ, eliminating r.
  8. Numerical solving methods should be a separate section. They are not a special case of el;imination.
  9. The penultimate sentence does not belong here.
  10. The last sentence also does not fit the topic "elimination". This has some relation to Ansatz methods.
  11. There are many more methods which sometimes may be succesful. A few examples would not hurt.

--LambiamTalk 15:26, 26 September 2006 (UTC)

  • Kay, I'm taking a look at it. I'll try and implement the requests you've listed here. --Brad Beattie (talk) 13:51, 19 November 2006 (UTC)

The cyclic method is mental math. Set up the difference of cross-products for the numerator of the value of each variable and another difference of cross-products for the denominator. Larry R. Holmgren 18:56, 23 March 2007 (UTC)

Although there is an example of a linear equation and a second degree equation (circle) other examples are systems of linear equations. Could we add a section on solving systems of quadratic equations. There are more steps but methods are similar, multiple substitutions and elimination of a variable. Larry R. Holmgren 04:03, 24 March 2007 (UTC)

Additionally, constrained systems of equations can be handled with a Lagrange multiple. I could add these three sections using equations of polynomials. The article on Lagrange multipliers does not have such a section.[1]

Yes, could someone else contribute systems of differential equations? An introduction would be good. The article on differential equations does not solve any, nor does it cover systems of equations. [2]Larry R. Holmgren 04:10, 24 March 2007 (UTC)

    1. "It is fine that it starts with an example, but then it should also give the solution for the example in the introductory paragraph, where perhaps something more elementary than the present example is better -- such as (x+2y = 7 & xy = 6) with solutions (x = 3 & y = 2) and (x = 4 & y = 1.5)."
      I did that.
    2. "The intro should not immediately talk about a geometric interpretation. ..."
      I did that.
Does anyone dispute my work? Just wondering ~user:orngjce223 how am I typing? 23:35, 12 October 2007 (UTC)

Merging Suggestion[edit]

I am proposing a merging of this article with part of Elementary algebra's info on Systems of Linear Equations and System of linear equations. Comments are appreciated and the explanation and discussion are being held here: Wikipedia talk:WikiProject Mathematics. (Quadrivium 23:17, 17 November 2006 (UTC))

Yes, the topics are the same and should be merged. Systems of Linear Equations is set up for a solution by determinants (matricies) which would have computer applications in statistics and econometrics and would be added to part 1.3 of this article, whereas Simultaneous equations is just a first year algebra topic, second semester. Larry R. Holmgren 19:06, 23 March 2007 (UTC)

No. As a schoolboy I've just come to this page to find out about these equations and it's been a great help and very well explained. I found it easily from Google. I don't want to get bogged down in any other topic. Leave it alone. 19:41, 11 April 2007 (UTC)

Is the article intended to be only about systems of simultaneous linear equations? An editor has just put a new tag "Merge with System of linear equations" on the article, and has next replaced all examples of non-linear equations in the article by linear equations. I find this a curious way of operation; it is like proposing to merge Religion with Buddhism and then proceeding, without awaiting the discussion, to erase all references to other belief systems than Buddhism from the Religion article.  --Lambiam 22:51, 19 March 2008 (UTC)
You're quite right. The distinction between the subjects is important, and there is so much to say about the linear case that a merged article would (appear to) have the general case as a footnote. I've been bold; I struck the merge tag (which was added 16 months after this talk section was started) and restored the worked-out non-linear example. (The linear example wasn't correctly solved anyway.) --Tardis (talk) 16:56, 17 April 2008 (UTC)
I propose a different action. I think this article, as well as System of linear equations, should be moved to System of equations, as this is the more common term used in academia (or for that matter, the term used at my school). Cheers, The Doctahedron, 23:09, 7 January 2012 (UTC)

Support with condition: System of equations already redirects here. I think that this article has to be rewritten into the following lines: After the lead, a description of the elementary methods. Then sections on the specialized and advanced methods, with template {main}: System of linear equations, system of polynomial equations, system of ordinary differential equations, ... (note that system of ordinary differential equation, which is grammatically incorrect, redirects to ordinary differential equation, which does not talk specifically on systems) The sections least-squares and matrices should either be suppressed or merged in the convenient specialized page. D.Lazard (talk) 11:57, 8 January 2012 (UTC)

  • Procedural oppose. The above template was added to a long-dead thread. It's not clear at all what is being voted on. I propose that this be marked as speedily declined, and a proper proposal and thread can be started. Sławomir Biały (talk) 12:44, 8 January 2012 (UTC)
    • The template was removed. If anyone really wants to rename this article, then feel free to start a discussion under a new heading but not above the current discussion. Vegaswikian (talk) 21:52, 8 January 2012 (UTC)

Cyclic Rule[edit]

I've reverted the addition of a section on this method from so-called "Vedic mathematics". Among the problems I see are the following (not necessarily in order of importance):

  • There is much wrong or confusing about the present article, which should be fixed before more confusing material is added.
  • This method is rather obscure and of unclear importance, and does not belong in an entry-level class article.
  • The description of the method is difficult to follow and unclear for me as a trained mathematician, and presumably fairly incomprehensible for the reader for whom the article is written.
  • The description does not set the context. To which kinds of systems of equations does it apply? Not to {5x − 4y + z + 8 = 0, 7x2 − 3y2 + 3z2 + 9 = 0, − 5y + z2 + 10 = 0}.
  • There is no mathematics markup of any kind.

Perhaps an improved version belongs in the article on Swami Bharati Krishna Tirtha's Vedic mathematics, or tucked away in a section on "Other solution methods" in System of linear equations.  --LambiamTalk 08:14, 27 March 2007 (UTC)


Simultaneous equations may well involve inequalities. This relates to the area of Linear Programming and Operations Research.--Billymac00 00:57, 15 September 2007 (UTC)


This section of the article is unintelligible. It needs to be written in much clearer language with simple worked examples. This explanation is useless to anyone who does not already know the method. —Preceding unsigned comment added by (talk) 15:44, 6 August 2009 (UTC)

Number of solutions[edit]

(Moving this discussion into a new section) D.Lazard (talk) 12:40, 16 May 2011 (UTC)

If the number of equations is the same as the number of variables, then probably (but not necessarily) the system is exactly solvable in the sense that the set of its solutions is infinite.

Should it really be infinite here, or finite? This would correspond more to my intuition of exactly solvable. —Preceding unsigned comment added by (talk) 19:30, 6 February 2010 (UTC)

Talk:Simultaneous equations/rewrite While the initial article about "Simultaneous equation" (since moved to "Simultaneous equations" and rewritten) was indeed impenetrable, statisticians do consider "Simultaneous equation models". gives 750 hits. AxelBoldt 23:00 20 Jun 2003 (UTC)

If there are fewer equations than variables, there may be infinitely many solutions; if there are more equations than variables, there may be no solution.

This is misleading: x=x, x+1=x+1 has infinitely many solutions, x+y=x+y+1 has none; reverted. - Patrick 02:09 21 Jun 2003 (UTC)

1) Yeah. The key words are "may have", I think. and 2) x+y=x+y+1 is not an equation.

Re:1) Disagree: A key word is "independent" - the many equations must be independent, otherwise one equation is just another equivalent form of a previous equation. For example, x=x and x+1=x+1 are not independent equations because if you add a constant to both sides of the second equation (which you are allowed to do in algebra and it doesn't change the equality) then you wind up with the first equation. E.g., add the constant -1 (minus one) to both sides, and the result is x=x. So x=x and x+1=x+1 are not two separate equations, they are really the same one. (talk) 17:54, 14 May 2011 (UTC)

Independent is clearly an important notion. Unfortunately, in the non linear case, testing independence is usually as difficult as solving. This matter is discussed in more details in article system of polynomial equations. Here under determined and over determined systems are considered, if the number of equations is respectively lower or higher than the number of unknowns. Over determined systems have generally no solution (inconsistent system), but may have some solutions if some hidden dependence relations occur. Under determined have generally an infinite number of complex solutions but may be inconsistent, like x+y = x+y+1 = 0. In the case of linear or polynomial systems, under determined systems have either an infinite number of complex solutions or no solution. But this is not true if only real solutions are considered or if the equations are not polynomial.
I agree that this article should be made more accurate, but it is difficult to do this, remaining at the low mathematical level needed for this article. Someone is willing to do this? D.Lazard (talk) 13:07, 16 May 2011 (UTC)


I cant see any citation,reference from reliable source except a webpage from Simon Bruce. I think Simultaneous equations or system of equations appear in Wikipedia thanks to WP:COMMONNAME, but in turn, seems to expand to article like Autonomous system, Nonlinear system, System of linear equations for example, see also Category:Systems theory and Portal:Systems science. The problem is the semi-mathematical and semi-pseudoscience properties of these article, we need discussion on whether they should be removed, rephrased or moved away from Mathematics Portal. The word system seems to be informal and ambiguous in Mathematics(WP:NATURALDIS) but it is quite common on other field(WP:COMMONNAME). Imo, this shouldn't be considered and can't be qualified as a project in Mathematics Portal due to the lack of verifiability or even notability within Mathematics community. In mathematics we often talk about family of sets and collection of objects, if there is some properties there we have tons load of algebraic structure like category, group. On the other hand, from the example articles given above, we can see they all suffer from lack of verifiability, and they lack section on precise definition on what a system is. They do mention the word system but evade the definition part, all wiki to this article which is in fact suffering from WP:SELFPUBLISH. -- (talk) 09:14, 30 March 2013 (UTC)

This article has multiple issues already described in this talk page, but not those of the preceding post: This article lacks of citations, but does not contain self published nor WP:original research; references for is content may be found in many places, inside (see, for example system of linear equations, elementary algebra and other targets of the links of this article) and outside WP. I do not know what is "semi-mathematical" and "semi-pseudoscience", but these term can not qualify this article, which, although elementary, is pure mathematics and thus pure science. "System" is ambiguous in mathematics (and not only there), but "system of equations" is not: this is a terminology that is well established since 19th century, much before the rise of "system theory" and "system science". Beside of the lack of citation, the main issue of this article is that (except for the lead), its content duplicates system of linear equations and, partly, elementary algebra and that it does not consider other types of systems of equations. IMO, the best way to solve these issues is to transform these article in a WP:DAB page. I'll be WP:bold and edit it in this way. D.Lazard (talk) 07:25, 3 April 2013 (UTC)