Question: A motel clerk counts his $1 and $10 bills at the end of a day. He finds that he has a total of 63 bills having a combined monetary value of $225. Find the number of bills of each denomination that he has.
1. **State the problem:**
A motel clerk has a total of $63$ bills consisting of $1$-dollar and $10$-dollar bills. The total amount of money is $225$. We need to find how many bills of each denomination he has.
2. **Define variables:**
Let $x$ be the number of $1$-dollar bills.
Let $y$ be the number of $10$-dollar bills.
3. **Write the system of equations:**
Since the total number of bills is $63$:
$$x + y = 63$$
Since the total value is $225$ dollars:
$$1 \cdot x + 10 \cdot y = 225$$
4. **Solve the system:**
From the first equation, express $x$ in terms of $y$:
$$x = 63 - y$$
Substitute into the second equation:
$$63 - y + 10y = 225$$
Simplify:
$$63 + 9y = 225$$
Subtract $63$ from both sides:
$$9y = 225 - 63$$
$$9y = 162$$
Divide both sides by $9$:
$$y = \frac{\cancel{9}162}{\cancel{9}} = 18$$
5. **Find $x$:**
$$x = 63 - y = 63 - 18 = 45$$
6. **Answer:**
The clerk has $45$ one-dollar bills and $18$ ten-dollar bills.