How to apply Implicit function theorem in multivariable calculus? I probably know that I have to use implicit function theorem in this problem but don't know how.  The following is the problem:


  The system of equations
  $3x+y-z+u^2 = 0$, 
  $x-y+2z+u=0$,
  $2x+2y-3z+2u=0$
  can be solved for $x, y,u$ in terms of $z$. Prove this.
Consider $$f(x,y,u,z) = \begin{bmatrix}
3x + y - z + u^2 \\ 
x - y + 2z + u\\ 
2x + 2y - 3z + 2u
\end{bmatrix}$$

The Jacobian is $$\left | \frac{\partial f}{\partial (x,y,u)} \right | = 2(4u-6).$$

So this is locally solvable in terms of $z$ as long as $u \neq 3/2.$
Start by adding the second and third equations, subtracting the sum from the first equation.  This yields $u^2$ - $u$ = 0, so $u$ = 0 or $u$ = 1 (or if you prefer, $u(z)$ = 0 or $u(z)$ = 1).  For each value of $u$, substitute back into the equations and solve the resulting linear system.  If you find one of these to be inconsistent, that lets you choose which $u$.