The problem is regarding GPS equations with four satellites with their location denoted as (A_{i}, B_{i}, C_{i}).

the transmission speed is approximately c (speed of light).

t_{i}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 - A_{1})^{2}+ (y - B_{1})^{2}+ (z - C_{1})^{2}= [c*(t_{1}- d)]^{2}

(x - A_{2})^{2}+ (y - B_{2})^{2}+ (z - C_{2})^{2}= [c*(t_{2}- d)]^{2 }(x - A_{3})^{2}+ (y - B_{3})^{2}+ (z - C_{3})^{2}= [c*(t_{3}- d)]^{2 }(x - A_{4})^{2}+ (y - B_{4})^{2}+ (z - C_{4})^{2}= [c*(t_{4}- 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?