SOLUTION: Find two numbers whose sum is 11 and whose difference between their squares is 33

We are given these two equalities

x   +  y  = 11,      (1)
x^2 - y^2 = 33.      (2)


Factor (2) in this way:

x^2 - y^2 = (x+y)*(x-y).  


Then (2) takes the form

(x+y)*(x-y) = 33.    (3)


Now replace (x+y) in (3) by 11,  based on (1). You will get

11*(x-y) = 33,    or

x - y = 3.          (4)


Thus equalities (1) and (2), taken together, are equivalent to

x + y = 11,         (5)
x - y = 3.          (6)


Now add  (5) and (6) (both sides).  You will get

2x = 11+ 3 = 14.  Hence,  x =  = 7.


Then from (5)  y = 11-x = 11 - 7 = 4.


Answer.  The two numbers are 7 and 4.