Systems of Linear Functions and their Reciprocals

Recently, I’ve been exploring the relationship between systems of linear functions and their reciprocals, namely: The following are the two methodologies I used in this investigation:

1)    Lo-tech method (longer way): Solve for all axis-intercepts, asymptotes and points of intersection for various examples.
2)    Higher-tech method (shorter way): Graph examples using software.

After sketching several examples, I noticed that at the points of intersection (poi) y = ±1.  This generalization was useful when calculating/checking the x-coordinate of the poi without the aid of graphing software.  After sketching the long way, similar results were confirmed through software. Calculating the poi algebraically: confirmed the observed coordinates for the poi: I wondered if this relationship could be extended to a system of functions such as: Plotting examples using graphing software didn’t yield any transparent relationship between the coefficients of the function and the poi coordinates. So, again, calculating the poi algebraically: resulted in the coordinates for the poi (with invariant point y = c^(1/2): Same same but different!