To find a linear demand equation for the number of guests $q$ per night as a function of the cover charge $p$, we can use the two given points from the problem. The data points are:

• When the cover charge was $9, there were 165 guests: $(p_1, q_1) = (9, 165)$
• When the cover charge was $13, there were 105 guests: $(p_2, q_2) = (13, 105)$

We assume the relationship between the number of guests and the price is linear, meaning the equation will take the form:

$q = mp + b$

Where:

• $m$ is the slope of the line,
• $b$ is the y-intercept.

Step 1: Find the slope $m$

The slope of the line between two points $(p_1, q_1)$ and $(p_2, q_2)$ is calculated by:

$m = \frac{q_2 - q_1}{p_2 - p_1}$

Substitute the given values:

$m = \frac{105 - 165}{13 - 9} = \frac{-60}{4} = -15$

Step 2: Find the y-intercept $b$

Now that we know the slope is $m = -15$, we use one of the points to solve for $b$. Let's use $(9, 165)$:

$165 = -15(9) + b$

Simplifying:

$165 = -135 + b$

$b = 165 + 135 = 300$

Step 3: Write the demand equation

Now that we have $m = -15$ and $b = 300$, the demand equation is:

$q = -15p + 300$

This is the linear demand equation showing the number of guests $q$ per night as a function of the cover charge $p$.

Would you like more details or have any questions? Here are some additional related questions to explore:

1. What does the slope of the demand equation represent?
2. What would happen to the number of guests if the cover charge increased to $15?
3. How many guests can you expect if you set the cover charge to $7?
4. What cover charge would attract exactly 150 guests?
5. How would the equation change if you lowered the cover charge to attract more guests?

Tip: When interpreting a linear demand equation, the slope tells you how much the demand (number of guests) decreases per dollar increase in the cover charge.