From #(1)# we have
#sqrt(2)x+sqrt(3)y = 0#
Dividing both sides by #sqrt(2)# gives us
#x + sqrt(3)/sqrt(2)y = 0" (*)"#
If we subtract #"(*)"# from #(2)# we obtain
#x+y-(x+sqrt(3)/sqrt(2)y) = sqrt(3)-sqrt(2) - 0#
#=> (1-sqrt(3)/sqrt(2))y = sqrt(3)-sqrt(2)#
#=> y = (sqrt(3)-sqrt(2))/(1-sqrt(3)/sqrt(2))=(sqrt(6)-2)/(sqrt(2)-sqrt(3))#
If we substitute the value we found for #y# back into #"(*)"# we get
#x + sqrt(3)/sqrt(2)*(sqrt(6)-2)/(sqrt(2)-sqrt(3)) = 0#
#=> x + (3sqrt(2)-2sqrt(3))/(2-sqrt(6)) = 0#
#=> x = -(3sqrt(2)-2sqrt(3))/(2-sqrt(6)) = (3sqrt(2)-2sqrt(3))/(sqrt(6)-2)#
Thus, we arrive at the solution
#{(x = (3sqrt(2)-2sqrt(3))/(sqrt(6)-2)), (y = (sqrt(6)-2)/(sqrt(2)-sqrt(3))):}#
