How do you isolate a variable in a function such as this

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Discussion Overview

The discussion revolves around the challenge of isolating a variable in a complex polynomial equation involving three variables: x, y, and z. Participants explore methods for expressing one variable, specifically z, in terms of the others, focusing on algebraic manipulation and substitution techniques.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the equation x^4+y^4+z^4+(4x^2)*(y^2)*(z^2)-34=0 and expresses difficulty in isolating z as a function of x and y.
  • Another participant suggests that it may not be possible to express one variable explicitly in terms of the other two.
  • A different approach is proposed, where one can substitute x^2 with w to transform the equation into a quadratic form, allowing for the use of the quadratic formula.
  • Further elaboration includes rewriting the equation to treat z as a polynomial, leading to a fourth-degree equation that can be simplified to a quadratic in terms of w=z^2.
  • One participant expresses appreciation for the insights shared, noting the utility of substituting squared variables.
  • Another participant describes their struggle with the quadratic equation and questions how to derive two functions from it, indicating confusion about the process.
  • A response questions the assumption that the function can be reduced to a specific form involving z as a function of x and y, suggesting that this is not generally applicable.
  • Further inquiry is made about whether there is a proof or general assumption regarding the reducibility of functions of three variables.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of isolating z in terms of x and y, with some suggesting it may not be possible while others explore methods to achieve this. The discussion remains unresolved regarding the generalizability of these approaches.

Contextual Notes

Participants acknowledge the complexity of the equation and the potential limitations of their methods, including the dependence on the specific structure of the polynomial and the assumptions made during substitution.

Storm Butler
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x^4+y^4+z^4+(4x^2)*(y^2)*(z^2)-34=0
I've tried all sorts of things factoring, algebraic manipulation ect. but i don't get how you could isolate a variable for a function like this i.e z= f(x,y).
 
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You may not be able express one variable explicitly in terms of the other two.
 
Everything seems to be raised to powers of two. Therefore for some variable say x, replace x^2 with w, and then find w using the quadratic equation.
 
Storm Butler said:
x^4+y^4+z^4+(4x^2)*(y^2)*(z^2)-34=0
I've tried all sorts of things factoring, algebraic manipulation ect. but i don't get how you could isolate a variable for a function like this i.e z= f(x,y).
If you want to solve for z (isolate z), treat it as a polynomial is z. For example, write it as [itex]z^4+ [4x^2y^2]z^2+ [x^4+ y^4- 34]= 0[/itex]
That's a fourth degree equation but, as John Chrieto suggests, since it has only even powers of z, letting [itex]w= z^2[/itex] gives [itex]w^2+ [4x^2y^2]w+ [x^4+ y^4- 34]= 0[/itex], a quadratic equation that can be solved using the quadratic formula- giving two solutions in terms of x and y. Then you can take the two square roots of each of those to give the four solutions to the fourth degree equation.
 
wow thank you guys you went well and beyond what i was thinking i didnt even think to substitute any of the squared variables and solve as a qudratic. Great insight/ intellegence.
 
ok so i tried doing that but i get stuck at this point: i did w^2+w*u+v=0 where w=z^2 and u and v are both equal to the separated "clump" functions. then i plug the variables into the quadratic equation and substitute and i get radical (x^4*(4y^4-1)-y^2-34)-2x^2*y^2 i don't get how you get two functions one in x and one in y.
 
Why do you think you should?

Very few functions of 3 variables can be reduce to "z= f(x)g(y)".
 
idk i just wasn't sure if i was doing something wrong or if i missed a step, also is your not the only one to say that I am curious is it just a general assumption or is there a proof for different classes of functions that tells if they can be reduced to z=f(x)y(x) ?
 

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