Tell me a little more about the pedagogical machine. Any friction anywhere ? Since you don't need the answer (it's already there), all you want is the "finding".
Your relevant equation doesn't mean a thing to me. X1 nor X3 are mentioned under 1. And if x1=x3 could help you find the acceleration if F = 0, it can hardly do so all on it's own. Any other relevant equations ?
Your "attempt" under 3 is a rephrasing of what you question under 2. CW and I both say yes: no way x1 - x3 can be anything but a constant -- at least if x is in the same direction as F ("sideways", so to speak; an assumption because the drawing does not reveal the direction of g). Not necessarily zero, but you could define coordinates (why don't you do that ?) such that yes, x1=x3.
As a student, you are forced to make a number of assumptions from context, brainwashing, tradition, or whatever. It is worth your effort to adapt an orderly way of dealing with exercises: list the problem statement, all given/known data; list all the relevant equations (if you can't, at least you should then mention what kind of relationship you are still missing). Link back to 1 and list the assumptions, clarify chosen coordinate systems etc.) Then make an attempt and tell us what you did.
Way you posted it now makes providing sensible help at the right level a nightmare to well-meaning people like me. So far, I all I can muster is 2. depends on choice of coordinates, 3. probably (assuming no significant gap around m3 turns that into a yes).
by jbunniii: All surfaces frictionless. And jbun deals with this Kleppner and Kolenkow exercise using the template. An example to you !
And you know, because I see you posting there with today's time stamp...