1. **State the problem:** We need to write a linear function that represents the total cost $y$ of a charity gala based on the number of attendees $x$. 2. **Identify the components:** The problem states there is a fixed cost of 150 for the ballroom and an additional 22.50 for each attendee. 3. **Write the formula:** The total cost $y$ is the sum of the fixed cost and the variable cost depending on attendees: $$y = \text{fixed cost} + (\text{cost per attendee}) \times x$$ 4. **Substitute the values:** $$y = 150 + 22.50x$$ 5. **Explain:** This means no matter how many people attend, the base cost is 150, and for each attendee, the cost increases by 22.50. 6. **Final answer:** $$y = 150 + 22.50x$$