Change of basis

I have a question regarding change-of-basis matrices. In our literature, the following equation:
(100a+100b,b) = a'(-1,2)+b'(1,2)
yields as a result:
a' = -(200a+199b)/4
b' = (200a+201b)/4

I do not understand how they get to this answer?
I'm asking this because I would like to apply it to the following question:

Consider two bases of R^2: B = {u1, u2} = {(1, -2), (3, -4)} and
L = {v1, v2} = {(1, 3), (3, 8)}. Find the change-of-basis matrix P from B to L.

From what I understand, you would have to find a,b,c,d so that:
(1,-2) = a(1,3) + b(3,8) and (3,-4) = c(1,3) + d(3,8)

But how do you go about this? I can see it with no calculations for easy answers, but for other matrices they are too complicated (like the one above where the answer would be-(200a+199b)/4). How do you get to such an answer?

Thanks for the help you may give me.

Uhm, sorry to have posted this question as my first post here. Because I have already figured it out myself. Sorry for the inconvenience.

The answer should be to solve ((1,3),(3,8))|(1,-2) as a simple gauss elimination right? Ahh I feel so dumb, sorry again.

I'm glad you figured it out on your own. It would helpful if you were now to mark your thread as "Solved". Should be under the option "Thread Tools" near the top-right corner of the OP.