Demand for widgets is a linear function of price.  If the price of Widgets is set to $20, 4200 widgets will be sold.  If the price is raised to $25, demand falls to 3600 widgets

To find the linear demand function, we need to determine the slope and the intercept of the linear equation $ Q = mP + b $, where $ Q $ is the quantity demanded, $ P $ is the price, $ m $ is the slope, and $ b $ is the y-intercept.

Given:
1. When $ P = 20 $, $ Q = 4200 $
2. When $ P = 25 $, $ Q = 3600 $

First, let's find the slope ($ m $):

\[
m = \frac{{Q_2 - Q_1}}{{P_2 - P_1}} = \frac{{3600 - 4200}}{{25 - 20}} = \frac{{-600}}{{5}} = -120
\]

So, the slope $ m $ is $-120$.

Next, we use one of the points to find the intercept ($ b $). Let's use the point $ (20, 4200) $:

\[
4200 = -120 \cdot 20 + b
\]

Solving for $ b $:

\[
4200 = -2400 + b
\]
\[
b = 4200 + 2400 = 6600
\]

Thus, the linear demand function is:

\[
Q = -120P + 6600
\]

### Summary:
The demand function for widgets is:
\[
Q = -120P + 6600
\]

Would you like more details or have any questions?

Here are some additional questions you might find interesting:
1. How would the demand function change if the slope were different?
2. What is the maximum price at which no widgets would be sold?
3. How would you determine the price elasticity of demand from this function?
4. How does the intercept of the demand function affect overall demand?
5. What assumptions are we making about the market in this linear model?

**Tip:** To understand the sensitivity of demand to price changes, consider calculating the price elasticity of demand, which measures the percentage change in quantity demanded resulting from a one percent change in price.

Learn how to calculate the linear demand function for widgets based on price changes. Understand the relationship between price and demand using real-world data points: $20 for 4200 widgets and $25 for 3600 widgets.

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