Inverting a system of equations technique?

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Discussion Overview

The discussion revolves around techniques for inverting a system of equations, specifically focusing on two equations that relate variables b, c, d, and e. The participants explore methods to express d and e in terms of b and c without directly using the quadratic formula, while considering the implications of changing the equations and the nature of the variables involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes to express e in terms of b and d using the equation e = b - d, and then substitute this into the second equation to derive a quadratic in d.
  • Another participant mentions the method of completing the square as a way to derive the quadratic formula, suggesting that it is a systematic approach to solving quadratic equations.
  • A participant expresses a desire to find a method for solving equations that can be generalized beyond quadratics, indicating an interest in systematic techniques applicable to various types of equations, including higher-order polynomials.
  • One participant emphasizes that the specific form of the equations is not crucial, as the variables are constants that could be complex, and they seek an inversion technique that may not have been considered.
  • A later reply introduces the concept of the Jacobian determinant and its role in determining the existence of an inverse transformation, noting that it is non-zero if and only if d is not equal to e.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and applicability of certain methods, such as the quadratic formula and completing the square. There is no consensus on a singular technique for inverting the equations, and multiple approaches are discussed without resolution.

Contextual Notes

Participants highlight the limitations of certain methods, such as the inability to use the quadratic formula directly, and the implications of the Abel-Ruffini theorem regarding the solvability of higher-degree polynomials. The discussion also touches on the need for numerical methods for complex equations.

Who May Find This Useful

Readers interested in mathematical problem-solving techniques, particularly those related to systems of equations, polynomial equations, and methods of inversion in algebra may find this discussion relevant.

Mugged
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I'm not entirely sure if this belongs in the linear algebra section, but here goes:

suppose I have two equations:

b = d + e

c = e*d

I have described b and c in terms of e and d. Is it possible to solve for e and d in terms of b and c? Can anyone provide some possible techniques?

Thank you
 
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Hi Mugged! :smile:

Use the first equation to get e = b - d,

then substitute for e into the second equation, to give you a quadratic equation in d :wink:
 
oh, yeah...forgot to mention. I can't use the quadratic equation directly.
 
Mugged said:
I can't use the quadratic equation directly.

what do you mean? :confused:
 
suppose I don't know the quadratic equation, how would i solve it then.

you see, the equations i put up come directly from a generic quadratic polynomial:

suppose I have an equation: x^2 + b*x + c = 0 and I try to put it in the form (x + e)*(x + d) = 0

the result is the equations above. So basically I'm trying to derive the quadratic formula without using it. Of course knowing the form of the quadratic formula helps you derive the forms for e and d, but I am just wondering is there a way to do it without knowing anything beforehand.
 
Typically one derives the quadratic formula using "completing the square".
 
Mugged said:
suppose I have an equation: x^2 + b*x + c = 0 …

you could write it x2 + bx + 1/4 b2 = -c - 1/4 b2,

which is the same as (x + b/2)2 = -c - 1/4 b2 :wink:

(that's completing the square)
 
oh come on...
 
It seems like you have a nail to put in as well as a hammer, but you don't for some reason want to use the hammer. Is there any reasoning behind this?
 
  • #10
Well, what I am really looking for is a systematic method for solving, so what I am really after is a nail gun.

If i change the equations in some manner, id like to still be able to solve them. Do you see what i mean? Its like if you have 2nd order ODE and solve it by finding the roots...that same root finding idea can be applied to a 3rd order ODE without radically changing the method.
 
  • #11
Mugged said:
Well, what I am really looking for is a systematic method for solving, so what I am really after is a nail gun.

If i change the equations in some manner, id like to still be able to solve them. Do you see what i mean? Its like if you have 2nd order ODE and solve it by finding the roots...that same root finding idea can be applied to a 3rd order ODE without radically changing the method.

Could you then express the generality in terms of the mathematical relationships.

As an example instead of it being a quadratic is it a polynomial? Maybe it's a transcendental function?

Instead of e being a constant could e = r(x,y,z)?

If you give us those things, then you will get a more specific answer to your question.

Also remember that techniques for general solutions like for example solving a general polynomial analytically are not at the very least unknown with the techniques we currently have in use.

So if you gave me say an arbitrary 10th degree polynomial and I had to find it's roots, I would probably in all likelihood have to use a computational numeric root finder to get an approximate answer, as opposed to using something that will spit out an analytic method.
 
  • #12
the type of equation i have used to come up with the equations in my first post is irrelevant. The letters b,c,d,e are all constants, possibly complex.

Im just wondering is there some inversion technique that can be used that i haven't considered?

and of course you'd have to use numerical for 10th degree. the abel-ruffini thm says no solution in radicals exists for degrees 5 or higher.
 
  • #13
The Jacobian:
[tex] \frac{\partial(b, c)}{\partial(d, e)} = \begin{vmatrix}<br /> 1 & 1 \\<br /> e & d<br /> \end{vmatrix} = d - e[/tex]
is non-zero if and only if [itex]d \neq e[/itex]. This is a necessary and sufficient condition for the existence of the inverse transformation.
 

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