I am trying to write a program to be able to "model" a Low Earth Orbit satellite's position at for a given time in the future. However I do not have a TLE for the desired satellite, I simply have a state-vector (Epoch and XYZ Position and Velocity). I am aware that this may not be a accurate projection for any significant amount of time, but this is all I have to work with.
What would the recommended steps be for working this out? Could I convert the state-vector into orbital elements somehow and then use those elements to calculate the position in respect to time? I am rather new to orbital mechanics so I apologize if this is a simple question. I have done some research and it appears the the orbital elements route seems like the route that most others have taken. However once I get those orbital elements, how do I calculate the satellites position?
Once I have an algorithm worked out I have other satellites with TLE's I plan to decompose into state-vectors and then compare the calculated state vector position with the calculated TLE position to ensure I am in the right ballpark.