The Chain Rule & Implicit Differentiation · Challenge problem

The plane $x + y + z = 2$ and the sphere $x^2 + y^2 + z^2 = 6$ meet in a curve that passes through $(1, 2, -1)$. Near this point, $y$ and $z$ are functions of $x$ along the curve. Find $\dfrac{dz}{dx}$ at $(1, 2, -1)$.