Hey dave, I'm not confused about anything but I would like to know when solving for c after you have unknowns a and b does it matter which equation you solve it in?
If you've solved correctly for a and b, then you'll get the same thing no matter which equation you use.
Similarly if you've eliminated c, then eliminated b from the two resulting equations, and then finally solved for a, it doesn't matter which of the two equations you substitute a into, you'll get the same result for b.
Once you've used your values of a and b to get c from one of the original equations, you should then substitute your a, b and c into another one of the original equations to check your work.
Another way of checking, of course, is to get your model y = a t^2 + b t + c and then to substitue your chosen t values. Each of your chosen t values should give you the corresponding y value.