I was reading a book on Calculus when I came across this:
$$\begin{cases} v+\ln(u)=xy \\ u+\ln(v)=x-y \\ \end{cases}$$
$$\begin{cases} \frac1u\frac{\partial u}{\partial x} +\frac{\partial v}{\partial x}=y \\ \frac1v\frac{\partial v} {\partial x} +\frac{\partial u}{\partial x}=1 \\ \end{cases}$$
$$\frac{\partial u}{\partial x} = \frac{ \begin{vmatrix} yu & u \\ v & 1 \\ \end{vmatrix} }{\begin{vmatrix} 1 & u \\ v & 1 \\ \end{vmatrix} }=\frac{u(y-v)}{1-uv}$$
I can get $\frac{\partial u}{\partial x}$ myself, but upon reaching the second system of equations I would have solved it with substitution. His use of determinants baffles me, and I have no form of Googling this. My questions are:
What is the name of this technique?
How were the elements of the matrix calculated?
A proof or a link to one would be a nice bonus as well.
This technique is called Cramer's rule.