Consider the system of equations
$7xy + 2y^2 - 9uv = 0$
$x^2 + 2xy - 6u^2 + 3v^2 = 0$
Notice that $(x, y, u, v) = (1, 1, 1, 1)$ is a solution of this system.
(a) Show that there is an $r > 0$ and a continuously differentiable function $G: B_r(1, 1) \rightarrow \mathbb{R}^2$
such that if
$(x - 1)^2 + (y - 1)^2 < r^2$,
$(u - 1)^2 + (v - 1)^2 < r^2$,
then $(x, y, u, v)$ is a solution of the system above if and only if $(u, v) = G(x, y)$.
(b) Find $DG(1, 1)$.
(c) If $G(x, y) = (G_1(x, y), G_2(x, y))$, what is $\frac{\partial G_1}{\partial y}$ at $(1, 1)$?
