The system is


y = 120 + 8x,    (1)
y = 20x,         (2)


where x is the number of boxes and y is the total cost.


To solve the system, notice that the left sides of the two equations are identical - so their right sides should be equal:


120 + 8x = 20x     (getting this equation is identical to substitution eq(1) into eq(2)).


Then 


120 = 20x - 8x

12x = 120  ====>  x = 120/12 = 10.


Answer.  10 boxes is break even.  Moving company is cheaper if the number of boxes is 11 or greater.


Then the total cost by the moving company would be 120+11*8 = 208 dollars,

                                  while by taxi it would be 11*20 = 220 dollars.