Consumers will demand 1365 items at a price of 2.75 per item. When the price per item increases to 4, the number of items demanded will decrease by 75 find the linear demand function p(x) for this item in slope intercept form

To find the linear demand function $ p(x) $, where $ p $ represents the price and $ x $ represents the number of items demanded, we will use the slope-intercept form of a line:

\[
p(x) = mx + b
\]

where:
- $ m $ is the slope,
- $ b $ is the y-intercept.

### Step 1: Identify the known points.
We have two points based on the problem statement:
- At $ x = 1365 $ items, $ p = 2.75 $ dollars (first point).
- At $ x = 1365 - 75 = 1290 $ items, $ p = 4 $ dollars (second point).

Thus, the points are $ (1365, 2.75) $ and $ (1290, 4) $.

### Step 2: Find the slope $ m $.
The formula for the slope between two points is:

\[
m = \frac{p_2 - p_1}{x_2 - x_1}
\]

Substitute the values from the two points:

\[
m = \frac{4 - 2.75}{1290 - 1365} = \frac{1.25}{-75} = -\frac{1}{60}
\]

So, the slope $ m = -\frac{1}{60} $.

### Step 3: Use the slope-intercept form to find $ b $.
Now, use one of the points (say $ (1365, 2.75) $) and the slope $ m = -\frac{1}{60} $ to find the y-intercept $ b $. Plug into the equation $ p = mx + b $:

\[
2.75 = -\frac{1}{60}(1365) + b
\]

Simplify:

\[
2.75 = -22.75 + b
\]

\[
b = 2.75 + 22.75 = 25.5
\]

### Step 4: Write the final demand function.
The linear demand function is:

\[
p(x) = -\frac{1}{60}x + 25.5
\]

This is the linear demand function in slope-intercept form.

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Would you like further details or have any questions? Here are some related questions to consider:

1. How does the slope $ m $ represent the relationship between price and demand?
2. What does the y-intercept $ b $ tell us in this context?
3. How would the function change if the demand decreased

Consumers will demand 1365 items at a price of 2.75 per item. When the price per item increases to 4, the number of items demanded will decrease by 75. Find the linear demand function p(x) for this item in slope-intercept form.

Learn how to find the linear demand function p(x) for an item when the price increases and demand decreases. In this example, consumers demand 1365 items at a price of $2.75, and demand drops by 75 items when the price rises to $4. The solution uses slope-intercept form, suitable for Grades 9-12.

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