Details of the GPS position calculation
The GPS calculation in the receiver uses four equations in the four
unknowns x, y, z, t_{c}, where x, y, z are the receiver’s
coordinates, and t_{c} is the time correction for the GPS
receiver’s clock. The four equations are:
where

c = speed of light (3 ´ 10^{8}
m/s)

t_{t,1}, t_{t,2}, t_{t,3}, t_{t,4}
= times that GPS satellites 1, 2, 3, and 4, respectively, transmitted
their signals (these times are provided to the receiver as part of the
information that is transmitted).

t_{r,1}, t_{r,2}, t_{r,3}, t_{r,4}
= times that the signals from GPS satellites 1, 2, 3, and 4, respectively,
are received (according to the inaccurate GPS receiver’s clock)

x_{1}, y_{1}, z_{1} = coordinates of GPS
satellite 1 (these coordinates are provided to the receiver as part of
the information that is transmitted); similar meaning for x_{2, }y_{2},
z_{2}, etc.
The receiver solves these equations simultaneously to determine x, y,
z, and t_{c}.