The problem is regarding GPS equations with four satellites with their location denoted as (Ai, Bi, Ci).
the transmission speed is approximately c (speed of light).
ti is the measured time intervals.
x, y and z is the receiver location, and d is the difference between the synchronized time on the four sattelite clocks and the earth-bound receiver clock.
(x - A1)2 + (y - B1)2 + (z - C1)2 = [c*(t1 - d)]2
(x - A2)2 + (y - B2)2 + (z - C2)2 = [c*(t2 - d)]2
(x - A3)2 + (y - B3)2 + (z - C3)2 = [c*(t3 - d)]2
(x - A4)2 + (y - B4)2 + (z - C4)2 = [c*(t4 - d)]2
I am to find the quadratic equation obtained from subtracting the last three equations from the first, and use those new three linear equations to eliminate x, y and z before finally substituting into any of the original equations. This is said to produce a quadratic equation in the single variable d.
I tried to do this by hand, and it got ugly, really ugly. Now I am stuck trying to figure out an easier way to accomplish this. Any ideas?