Talk:Simultaneous equations

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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". http://www.google.com/search?q=site%3A.edu+simultaneous+equation+endogenous+Hausman 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.

Contents

[edit] Material moved out of another page

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.

==References==

[edit] Expert needed

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)

[edit] Merging Suggestion

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. 86.31.78.115 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)

[edit] Cyclic Rule

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)

[edit] Inequalities

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