The problem is asking for a linear equation that models the relationship between the price of a device and the number of devices sold per month. Let's break it down step by step.

Information Given:

• At a price of $5, the company sells 1050 devices per month.
• At a price of $9, the company sells 900 devices per month.
• The relationship between price and sales is linear.

We need to find the number of devices sold when the price is $12.

Step 1: Write the relationship as a linear equation

Let $p$ represent the price of the device, and $S(p)$ represent the number of devices sold per month. Since the relationship is linear, we can model it with a line: $S(p) = mp + b$ where $m$ is the slope and $b$ is the y-intercept.

Step 2: Calculate the slope $m$

We are given two points:

• $(5, 1050)$ — when the price is $5, the company sells 1050 devices.
• $(9, 900)$ — when the price is $9, the company sells 900 devices.

The slope $m$ is calculated using the formula for the slope of a line: $m = \frac{S(p_2) - S(p_1)}{p_2 - p_1} = \frac{900 - 1050}{9 - 5} = \frac{-150}{4} = -37.5$ So, $m = -37.5$.

Step 3: Find the y-intercept $b$

Now, use one of the points, say $(5, 1050)$, and substitute it into the equation $S(p) = mp + b$ to solve for $b$: $1050 = -37.5(5) + b$ $1050 = -187.5 + b$ $b = 1050 + 187.5 = 1237.5$

Step 4: Write the equation

The linear equation that models the number of devices sold is: $S(p) = -37.5p + 1237.5$

Step 5: Calculate sales when the price is $12

Substitute $p = 12$ into the equation: $S(12) = -37.5(12) + 1237.5$ $S(12) = -450 + 1237.5 = 787.5$

Step 6: Round to the nearest whole number

The number of devices sold when the price is $12 is approximately 788.

Thus, the correct answer is: $\boxed{D. 788 \text{ devices}}$

Would you like more details or have any further questions? Here are some related questions to expand on this problem:

1. What would happen if the price increased to $15? How many devices would be sold?
2. How would the equation change if the relationship between price and sales were not linear?
3. How can you interpret the slope of -37.5 in terms of the company's pricing strategy?
4. What is the significance of the y-intercept in this model?
5. Can you think of other factors besides price that could influence the number of devices sold?

Tip: When dealing with linear models, the slope represents the rate of change between the two variables. In this case, for every $1 increase in price, 37.5 fewer devices are sold.