Rather than fooling with algebra let's examine the principle with some drawings. These drawings will be a lot easier to understand if we keep them two-dimensional.

Of course, GPS is a 3-dimensional system but the principles will work just fine in two dimensions. Just remember that in the real 3-dimensional system we need to add one more measurement.

Also we've been talking about satellite ranges in terms of distance but since these ranges are just calculated from time let's simplify things by talking about ranges as times.

OK, the object here is to see how an extra satellite range can be used to determine if our receivers clocks are out of sync with universal time.

Let's say that in reality our position is four seconds from satellite A and six seconds from satellite B. Those two ranges cross and the intersection is our position (remember we're only working in 2D).