Given the system of equations :
F(x,y,z,u,v,w)=14x + 2y - 8z^2 - 3u + 9v^2 - 68 = 0
G(x,y,z,u,v,w)=28x + 4y + 5z + 5u^2 + 5v 10w - 528 = 0
H(x,y,z,u,v,w)=2x + 4z + 4u^2 + 8w - 274 = 0
**
Determine whether the system of equations is solvable for u , v , w as functions
of x , y , z near the point P where (x, y, z) = (7,-2,-4) , and (u, v,w) = (-7, 3, 10).
What is the condition that will guarantee the system of equations is solvable for
x , y , z as functions of u , v , w ?

Question

Given the system of equations :
F(x,y,z,u,v,w)=14x + 2y - 8z^2 - 3u + 9v^2 - 68 = 0
G(x,y,z,u,v,w)=28x + 4y + 5z + 5u^2 + 5v  10w - 528 = 0
H(x,y,z,u,v,w)=2x + 4z + 4u^2 + 8w - 274 = 0
**
Determine whether the system of equations is solvable for u , v , w as functions
of x , y , z near the point P where (x, y, z) = (7,-2,-4) , and (u, v,w) = (-7, 3, 10).
What is the condition that will guarantee the system of equations is solvable for
x , y , z as functions of u , v , w ?

jamesj · Answer

Are you sure about the first equation with the z^2 term? 

If it is just z then ... you have three inhomogeneous linear equations in x, y, z equal to constant terms equal to functions in u, v, w.

So the question of solubility is simply a question of elementary linear algebra in disguise.  Does the matrix of coefficients of x, y, z have the desired properties?

anonymous · Answer

I'm not sure I follow on what you are trying to say... all I know is I typed the equations right. And we are using Jacobians or partial differentation. I'm not sure if you can use the same method using linear algebra, I wouldnt know how to do it. Please elaborate if you know.

jamesj · Answer

Do you have an example of problem like, with perhaps 4 variables--such as x,y; v,w--that you can show us?