When we write a vector, v, in terms of the basis (w1, w2), we are writing it as v= aw1+ bw2 for some numbers a and b. You want x= -1w1+ 2w2= (5, 6) and y= 2w1- w2= (-1, 0).

If you let w1= (a, b) and w2= (c, d) then that says that -1(a, b)+ 2(c, d)= (-a+2c, -b+ 2d)= (5, 6) and 2(a, b)- (c, d)= (2a- c, 2b- d)= -1, 0). That gives you four equations for a, b, c, and d. (Actually, two equations for a and c and two equations for b and d.)