This article explains how to do trilateration step by step.
I need to extend this process to 3-D. As far as I know, I need four distance measurements in order to calculate a fifth point's coordinates.
Could you help me to extend trilateration to 3-D?
P.S. Please do not answer with "solve the linear system" I need step by step calculations.
The article you cited is about locating a point in 3-D space. Given three given non-collinear points and the distance from each of these to a point-to-be-found, $P$, the article enables you to narrow down the possible locations of $P$ to at most two distinct points. If there are two such points, then given a fourth known point not coplanar with the other three, the two candidates for $P$ will be at different distances from that fourth known point, and knowing the distance from that point to $P$, you can determine which candidate is correct.