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.