1. **Stating the problem:** A hotel has 9 rooms, each with either 3 or 4 beds. A group of 30 people occupies all rooms completely. We need to find the smallest number of rooms that have 4 beds. 2. **Setting variables:** Let $x$ be the number of rooms with 4 beds, and $y$ be the number of rooms with 3 beds. 3. **Formulating equations:** Since there are 9 rooms total: $$x + y = 9$$ The total number of beds must equal 30 (since all people occupy all beds): $$4x + 3y = 30$$ 4. **Solving the system:** From the first equation, express $y$: $$y = 9 - x$$ Substitute into the second equation: $$4x + 3(9 - x) = 30$$ Simplify: $$4x + 27 - 3x = 30$$ $$x + 27 = 30$$ $$x = 30 - 27$$ $$x = 3$$ 5. **Check $y$ value:** $$y = 9 - 3 = 6$$ 6. **Interpretation:** The smallest number of rooms with 4 beds is $3$. **Final answer:** 3