I need some help with the completing the task given in the last sentence of this old exam question.
"The system
$$
x+y+z&=6 \\
x^2+y^2+z^2&=14
$$
is satisfied at the point $(1,2,3)$. Show that $x$ and $y$ can be solved in a neighborhood of $(1,2,3)$ as a function of $z$. Calculate also $x'(3)$ and $y'(3)$, where $x$ and $y$ are regarded as functions of $z$.".
Any clues?