Can someone explain to me how to solve a problem like:

27x + 9
(pretend its a line)
3

27x + 9 

     3

Break down the fraction. 3 is the common denominator, so it is the denominator of both numerators.

27x/3 + 9/3 = 0
9x + 3 = 0 
9x = -3
x = -3/9  simplified to x = -1/3

Or simply multiply the fraction with 3/1 to remove the denominator, which is 3.

27x + 9         3            27x + 9
  x    =    or 27x + 9 = 0 ; look for x.
     3              1                 1

To check: Substitute x by its value: 
(27x + 9) / 3 = 0 ⇒ ((27 * -1/3) + 9) / 3 = 0 
⇒ (-9 + 9)/3 = 0 ⇒ 0/3 = 0 ⇒ 0 = 0