Like you I don't know Latex so see the attachment
By the way you're region is bounded ny x = 0 , y= 0 and y = 1-x
Its all in the attachment
In an e-mail I got you said you had difficulty finding x and y in terms of u and v but it seems you figured it out
In case you haven't there are 2 ways
1. Using matrices --see attachment
2.Brute force and ignorance
u =x+y
v = -2x+y
subtract the 2 eqns --this eliminates y solve for x
use the result in either eqn to find y