One approach is to isolate y in the bottom formula, then fill it in to the top one:
x-y=-3 => y = 3+x
so substitute 3+x for y in the top formula (watch the signs) and simplify
x² - 3 - x = 3 => x² -x -6 = 0 => (x+2)(x-3)=0
so x=-2 or x=3.
for x=-2, y=1
for x=3, y=6
These are the solutions.
