Talk:Simultaneous equations

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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.

[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)

[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))

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