Consider the system of 2 equations 

y = ,              (1)

y - (-4) = m(x-4).       (2)



It is the same, as the system

y = ,              (1)

y + 4 = m(x-4).          (2)



Find  the slope "m" under the condition that the system has a UNIQUE solution (x,y) = (4,-4).


For it, substitute expression for y,  y = m*(x-4)-4  into equation (1). You will get

m*(x-4) + 4 = .       (*)


Introduce new variable  u = .  Then the equation (*) takes the form

 = ,

 = ,

 = 0.             (**)


The condition that the equation (**) has a unique solution in "u"  means that the discriminant of the equation (**) is equal to zero.


Write the discriminant of the quadratic equation (**) and equate it to zero:

d =  =  = 16 - 16m  ====>

the equation d = 0  becomes  16 - 16m = 0,  which implies  16m = 16  ====>  m = 1.


Answer.  The slope under the question is  m= 1,
      
         found without using derivatives.