Why does $\frac{\frac yx - \frac xy}{\frac1x - \frac1y} = x + y $ and $\frac{\frac xy - \frac yx}{\frac1x - \frac1y} = -(x + y)$? As the titles states: Why does $\frac{\frac yx - \frac xy}{\frac1x - \frac1y} = x + y $ and $\frac{\frac xy - \frac yx}{\frac1x - \frac1y} = -(x + y)$?
I am only able to find $\frac{y^2 - x^2}{y - x}$ and $\frac{x^2 - y^2}{y - x}$ respectively.
How I have done it so far: 
$$\frac{\frac yx - \frac xy}{\frac 1x - \frac 1y} =
    \frac{ \frac{y^2 - x^2}{xy} }{ \frac {y - x}{xy}} = 
    \frac{y^2 - x^2}{y - x}$$
$$\frac{\frac xy - \frac yx}{\frac 1x - \frac 1y} =
    \frac{ \frac{x^2 - y^2}{xy}}{\frac {y - x}{xy}} = 
    \frac{x^2 - y^2}{y - x}$$
I reached $x + y$ and $-(x + y)$ respectively by using a sort of trial-and-error and looking at bigger/smaller than (each other) and negative/positive arbitrary values for $x $ and $y$ before I confirmed it on wolfram alpha. I can't wrap my head around how I would actually prove it though as I barely understand my own reasoning other than that it works. I got to $ x + y$ and $-(x + y)$ using common sense rather than math..
I hope I'm making sense and appreciate any and all help I can get!

Notice that both numerators are differences of squares:
$$
\frac{y^2 - x^2}{y - x} = \frac{(y - x)(y + x)}{y - x} = y + x
$$
and likewise:
$$
\frac{x^2 - y^2}{y - x} = \frac{(x - y)(x + y)}{y - x} = \frac{-(y - x)(x + y)}{y - x} = -(x + y)
$$
