Check the problem again. A piece of information is missing.  
There are 5 unknowns, but there is only enough information 
to get 4 equations.  That's called an "underdetermined" system.
(I'm assuming you're not taking linear algebra, or are you?
I'll assume you aren't.)  We can at best get some of times, 
and we may not get any! But let's see how far we can get
without the missing piece of information.

Let The answers be:
It takes A "a" hours to complete the task alone.
It takes B "b" hours to complete the task alone.
It takes C "c" hours to complete the task alone.
It takes D "d" hours to complete the task alone.
It takes E "e" hours to complete the task alone.

Then 
A's working rate is 1 task per "a" hours or  tasks/hour
B's working rate is 1 task per "b" hours or  tasks/hour
C's working rate is 1 task per "c" hours or  tasks/hour
D's working rate is 1 task per "d" hours or  tasks/hour
E's working rate is 1 task per "e" hours or  tasks/hour

So their combined working rate is  1 task per "2" hours or  tasks/hour

  

So

  

4 hours and 48 minutes is  =  = 

So their combined working rate is 1 task per  hours or  or  tasks/hour

  

  

So their combined working rate is 1 task per 4 hours or  tasks/hour.

  

  

3 hours and 12 minutes is  =  = 

So their combined working rate is 1 task per  hours or  or  tasks/hour

 

  

So we have the system of 4 equations in 5 unknowns;  



We consider the reciprocals to be the variables and
make this 4×6 augmented matrix:



Using a TI graphing calculator I get the "rref":



The third row translates as

, so c=16 and it takes C 16 hours to complete the task

So we substitute c=16 in the system



and simplify



We'll put it in a 4×5 augmented matrix.  But I'm quite 
sure that will be a singular matrix.



And get the "rref"



See? The bottom row is all 0's.

So the only answer we can get without the missing piece of
information is that C can complete the task in 16 hours.

Edwin