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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.

If a large company produces and sells its new device for $5 each, it can sell 1050 per month, and if it sells the same device for $9, it will sell 900 per month. Assuming the relationship between price and sales is linear, which of the following equations models the monthly sales of devices to the nearest whole number if the price is $12? A. 572 devices B. 612 devices C. 701 devices D. 788 devices

Explore how to model the monthly sales of devices using a linear equation. Given price points and corresponding sales data, learn how to calculate the slope and predict sales at different price levels. Ideal for students in Grades 9-11.

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