Show that there are positive numbers $p$ and $q$ and unique functions $u$ and $v$ from the interval $(-1-p, -1+p)$ into the interval $(1-q, 1+q)$ satisfying $$xe^{u(x)} +u(x)e^{v(x)}=0=xe^{v(x)} +v(x)e^{u(x)}$$ for all $x$ in the interval $(-1-p, -1+p)$ with $u(-1)=1=v(-1)$.
Progress
The first step is to take the Jacobian, which I did, but that didn't do much for me. The Jacobian was $$(x^{2}+1 -uv)e^{u+v}+x(e^{2u}+e^{2v})$$
Taking derivatives of both equations with respect to $u,v$ and forming the Jacobian led to $$(x^{2}+1 -uv)e^{u+v}+x(e^{2u}+e^{2v})$$ We are told to look at a neighborhood of the point $x=-1$, $u=1$, $v=1$. At this point, the Jacobian evaluates to $-e^{2}$, which is nonzero. Therefore, the Implicit Function theorem applies: there is a neighborhood of $-1$ in which $u(x)$ and $v(x)$ are smooth functions with $u(-1)=1$, $v(-1)=1$.