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Lately I've been running into certain types of problems in my algebra class which are proving to be time-consuming for me, so I thought I'd express my difficulties and hope someone could offer some advice.

Given two vectors {(1,1), (1,2)}, I have to prove that they span the cartesian plane. So the way this is done is to express an arbitrary vector in the plane in terms of these vectors. Simple enough, but it's proving to take more time than it should for me.

So for this specific example, my professor expressed an arbitrary vector by "picking" the following coefficients:

(x,y) = (2x-y)(1,1) + (y-x)(1,2).

I know he just doesn't pick something that works and I know I shouldn't be expected to get a solution immediately. But how would I approach these type of mechanical problems, to find coefficients that work? It would be helpful to have an approach in mind and some way to achieve it, because whenever I run into these problems, it takes a long time (when I should be moving onto more important concepts).

Thanks in advance.