Without using algebra:

Since there were twice as many boys as girls, think of dividing the
boys into two equal groups, and letting all the girls be a third group.
Then each of those 3 groups would have the same number of people. 

Then if 1 of those groups of boys left the room, there would be just two
equal groups, one of boys and 1 or girls.  

So the 5 boys that left must have been one of those groups.  So at first,
there were 10 boys and 5 girls in the room.

    
Using algebra:



Substitute (2g) for b in the second equation:





Substitute (5) for g in 



10 boys and 5 girls in the room at first.

Edwin