GPS, or Global Positioning System is a network of satellites in orbit above the surface of the Earth. By the satellites constantly transmitting their exact position and on-board time through radio waves, GPS receivers can calculate their exact position knowing the position of four other satellites by first using the basic formula distance= speed×time (as the speed of light is constant at ~300,000,000 m/s {this gets complicated, but this is good enough in context of my actual question}), and then using trilateration to compute the receiver's exact position.On the satellites, highly accurate time is maintained using atomic clocks.
However, GPS receivers themselves do not have atomic clocks on-board, so how do they maintain accurate time for this whole process? I've read in a few places that they use some math to calculate their time using the positions of four other satellites, but I couldn't find a system of equations to describe this method.
A GPS receiver does not need to keep time, in fact the purpose of many GPS receivers is simply to get time from the GPS system. You can think of the data stream from each satellite as giving the position of the satellite and the time corresponding to that position. When the receiver gets that data it knows it is on a sphere around the satellite position with radius the light travel time from that position. That is a sphere that is expanding in time with the speed of light. It is an equation in four variables, the $x,y,z$ coordinates of the receiver and the actual time of reception saying the time from emission to receipt is the speed of light. If you receive the data from four satellites you have four equations in four unknowns. They can be solved simultaneously. As the equations are quadratic, there could be multiple solutions, but no more than sixteen. The ambiguity can be resolved by getting more samples from the same satellites, by getting data from another satellite, or by knowing approximately where the receiver is and choosing the closest solution.