Observation equations vs. Condition equations.
I don't know if the quote above was intended to suggest the process being discussed needs condition equations AND it has been way too long since I have done one of these long hand, so I may be mistaken in two instances. But, I don't think the method the professor is requiring needs/uses condition equations. The variation of coordinates uses the observation equations to develop the matrix that is reduced to find a recommended change in north and east coordinate value to minimize the errors in the system of equations. That recommended change is applied to the seed coordinates, replacing the first approximation of coordinate values and then the matrix math is repeated (rinse, lather, repeat, until the error values reach an accepted threshold). No need to include a condition as the results are based only on the observations and may or may not fit geometric "knowns" perfectly.
At least that is my recollection from a very long time ago when we would use a program called TK Solver to run the iterations. I think TK Solver is still around, but I am talking about when the same software company offered the (at the time) premiere spreadsheet software - Lotus 1-2-3 (not to be confused with the knockoff software "As Easy As..." which certainly made good use of an old saying).