Let x be the number of $1 bills and y be the number of $10 bills.
We know that:
x + y = 59 (the total number of bills)
x + 10y = 239 (the total monetary value of the bills)
We can use the first equation to solve for one of the variables in terms of the other. For example:
x = 59 - y
We can substitute this expression for x into the second equation:
(59 - y) + 10y = 239
59 + 9y = 239
9y = 180
y = 20 (the number of $10 bills)
We can use this value of y to find the number of $1 bills by substituting it back into the first equation:
x + y = 59
x + 20 = 59
x = 39 (the number of $1 bills)
Therefore, the motel clerk has 39 $1 bills and 20 $10 bills.