Without algebra (you can do it in your head this way):

If he had only $5 bills he would have 1000÷5 or 200 bills.  But since
he had only 125 bills, then 200-125 or 75 of them had to be worth twice 
as much as a $5.  So he had 75 tens and 125-75 = 50 fives. 

With algebra  (which is the way your teacher wants you to do it)

The other tutor used two unknowns, but it can be done with only 1 unknown.

Let x = the number of $5 bills.
Then the number of $10 is 125-x

So

            

            5·x + 10·(125-x) = 1000
             5x + 1250 - 10x = 1000
                         -5x = -250
                           x = 50

Edwin