Jenny, there's no reason why you can't use it, given that it's pretty much "derived" by common definitions.


Now just add them:

t)
Which is a common formula. There you go, it's not hard to derive and if you think about it, it's common sense because what you "add" to the initial velocity is what you "subtract" from the final velocity, if that makes sense.
final and initial velocity are totally different. I'm unsure if we can use it without proving
Not really, they're not "totally different" - remember that we are talking about rectilinear motion with constant acceleration, meaning that the final velocity is dependent upon the initial velocity. So they are related.
And lastly, Jenny, Physics is completely different to maths - in Physics, you're allowed to (and some would even say encouraged to) take shortcuts and use quick formulas, because marks are awarded for the correct answer in the box, i.e. if you get a correct answer in the box, you will be awarded full marks, regardless of your working.