Let x = the number of the 5-dollar bills, and
let y = the number of the 10-dollar bills.


Then from the condition you have these two equations

     x -   y =  19      (1)    ("19 more five dollar bills than ten dollar bills")
   5x  + 10y = 530      (2)    ("the total amount of money in the cash register is $530")


From equation (1) express x = 19 + y  and substitute it into equation (2).
You will get


    5*(19+y) + 10y = 530,    (3)

    95 + 5y + 10y = 530,

    15y = 530 - 95 = 435  

      y =  = 29.


Answer.  29 10-dollar bills  and  29 + 19 = 48 5-dollar bills.


Check.   10*29 + 5*48 = 530 dollars total.   ! Correct !

Let y = the number of the 10-dollar bills.

Then the number of the 5-dollar bills is (y+19), according to the condition.


10-dollar bills contribute  10*y     dollars to the total.

5-dollar  bills contribute  5*(y+19) dollars to the total.


So the money equation for the total is


    10y + 5*(y+19) = 530.


It is THE SAME EQUATION as  equation (3) in the Solution 1 above.

You can solve it by the same way, and you will get the same answer.