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

[Storm Butler]

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).

 

[JG89]

You may not be able express one variable explicitly in terms of the other two.

 

[John Creighto]

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.

 

[HallsofIvy]

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 $z^4+ [4x^2y^2]z^2+ [x^4+ y^4- 34]= 0$
That's a fourth degree equation but, as John Chrieto suggests, since it has only even powers of z, letting $w= z^2$ gives $w^2+ [4x^2y^2]w+ [x^4+ y^4- 34]= 0$, 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.

 

[Storm Butler]

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.

 

[Storm Butler]

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.

 

[HallsofIvy]

Why do you think you should?

Very few functions of 3 variables can be reduce to "z= f(x)g(y)".

 

[Storm Butler]

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) ?