Here is how you might write the laws out mathematically correctly:
Law 1: p1v1 = p2v2 only when t1 = t2
Law 2: v1/t1 = v2/t2 only when p1 = p2
Law 3: p1/t1 = p2/t2 only when v1 = v2
Now when you try to combine say Law 1 and Law 2, you get:
(from Law 1) v1 = p2v2/p1 only when t1 = t2
(substituting into Law 2) p2v2/p1t1 = v2/t2 only when t1 = t2 and p1 =
Continue this as you probably did and get a formula with a t2 where you expected a t1. But since t1 = t2 in the restrictions of the formula by then, this is true.
Bottom line is that you can not just algebraically combine these three laws given that they each have different restrictions. If you do you will just get a combined formula where the restrictions that go with it make it meaningless.
If you want to get to the combined gas law I suggest looking at it in a more simple manner.
Note that Law 1 can be written as p1v1/t1 = p2v2/t2 when t1 = t2
Note that Law 2 can be written as p1v1/t1 = p2v2/t2 when p1 = p2
Note that Law 3 can be written as p1v1/t1 = p2v2/t2 when v1 = v2
So all three laws can be written as the combined gas law using their restrictions. But if Law 2 shows that the temperature does not need to stay constant for the formula to be correct then we can take that restriction out. You can similarly take out the other restrictions and get the combined gas law with no restrictions.
If you want to see an algebraic way to derive it go tohttp://dbhs.wvusd.k12.ca.us/webdocs/GasLaw/Gas-Combined.html
but note that this is a gimmick method. They do not take into account the restrictions, opting to just use methods that get the desired result.