Question 445162
Let h,d, and l be the number of hours it takes for each to separately paint a room.

So here's the trick: the efficiency of each is (1 room) / (number of hours it takes). In essence, it's the fraction of the room painted in 1 hour.

{{{(1/h) + (1/d) + (1/l) = 1/4}}}
{{{(1/h) + (1/l) = 1/6}}}
{{{(1/d) + (1/l) = 1/7}}}
add the bottom two equations to get
{{{(1/h) + (2/l) + (1/d) = (13/42)}}}
{{{(1/h) + (1/l) + (1/d) = (13/42) - (1/l)}}}
Recall the first equation and make a substitution.
{{{(1/4) = (13/42) - (1/l)}}}
{{{(13/42)-(1/4) = (1/l)}}}
{{{(26/84)-(21/84) = (5/84) = (1/l)}}}
{{{5l = 84}}}
{{{l = 84/5}}}} or 16.8 hours or 16 4/5 hours
Plug l = 84/5 into the 2nd equation to get 
{{{1/h + 5/84 = 1/6}}}
{{{1/h = 5/84 = 14/84}}}
{{{1/h = 9/84}}}
{{{h = 84/9}}} or 9.33333 hours or 9 1/3 hours
Plug l = 84/5 into the 3rd equation to get
{{{1/d + 5/84 = 1/7}}}
{{{1/d = 7/84}}}
{{{d = 84/7}}} or 12 hours.