Change of Variables, how do I determine u and v ? For example how were the value

Question

Change of Variables, how do I determine u and v ? For example how were the values u=x-2y and v=x+y determined in the following example. And how do you come up with x = (1/3)(u+2v) and y= (1/3)(-u+v)????? Please help me, if you could explain it in as much detail as possible I would greatly appreciate it!

SPECIFICALLY: How do I figure out what X and Y equal in terms of U and V? I do not understand how it was arrived upon that u=x-2y and v=x+y . It must be incredibly simple and I am just missing it because I am having the hardest time finding an explaination for this.

Explanation / Answer

Hey its pretty straight forward
first we take u and v based on given conditions, here we have x-2y=0 and x+y=1 and remaining to eqns
so we can use any two of them for our purpose here we took these 2 for simlipcity
So we take the values u=x-2y and v=x+y and
teh solving for x and y we get x = (1/3)(u+2v) and y= (1/3)(-u+v).
I hope this helps!

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