Problem in understanding Blakley's Secret Sharing Scheme
An easier way of explaining it is here: http://www.demoivre.org/courses/CIS628/chapter15.pdf

So for a point $(x_0,y_0,z_0)$ we set $x_0$ as the secret and then randomly choose $y_0$ and $z_0\pmod{P}$.

Now we  generate our plane to distribute to the participants:
we pick two random integers $a$ and $b$, then we set

$$C = z_0 - ax_0 -by_0 \pmod{P}$$

we now have our equation for the plane, such that:

$$z = ax + by + c$$

to reassemble the secret we use matrix manipulation and it is simply:

$$a_i x + b_i y - z \equiv -c_i \pmod{P} $$

scroll down to page 12 on that link for a really good example