In particular, historical data shows that 9000 shirts can be sold at a price of $67 , while 16000 shirts can be sold at a price of $4 . Give a linear equation in the form p = mn + b that gives the price p they can charge for shirts.

To find the linear equation $ p = mn + b $, where $ p $ is the price and $ n $ is the number of shirts sold, we need to use the two data points provided:

1. When 9000 shirts are sold, the price is $67.
2. When 16000 shirts are sold, the price is $4.

We will treat these as two points on a line: $ (n_1, p_1) = (9000, 67) $ and $ (n_2, p_2) = (16000, 4) $.

### Step 1: Find the slope $ m $
The slope of a line $ m $ is calculated using the formula:
\[
m = \frac{p_2 - p_1}{n_2 - n_1}
\]
Substituting the values:
\[
m = \frac{4 - 67}{16000 - 9000} = \frac{-63}{7000} = -0.009
\]
So, the slope $ m $ is $ -0.009 $.

### Step 2: Use the point-slope form to find the intercept $ b $
The point-slope form of a linear equation is:
\[
p - p_1 = m(n - n_1)
\]
Using the point $ (9000, 67) $ and $ m = -0.009 $:
\[
p - 67 = -0.009(n - 9000)
\]
Expanding this equation:
\[
p - 67 = -0.009n + 81
\]
Solving for $ p $:
\[
p = -0.009n + 81 + 67
\]
\[
p = -0.009n + 148
\]

### Final Equation:
The linear equation is:
\[
p = -0.009n + 148
\]
This equation gives the price $ p $ based on the number of shirts $ n $.

Would you like more details or clarifications on this? Here are some related questions you might explore:

1. How would the equation change if the price for 20,000 shirts was $3?
2. What does the slope $ m = -0.009 $ tell us about the relationship between price and number of shirts?
3. How can you use this equation to predict the price for selling 12,000 shirts?
4. Can you graph this equation and interpret the intercept?
5. How would the linear equation differ if it were a quadratic relationship instead of linear?

**Tip:** The slope of a linear equation shows how sensitive the price is to changes in the quantity sold. A negative slope indicates that the price decreases as more shirts are sold.

In particular, historical data shows that 9000 shirts can be sold at a price of $67, while 16000 shirts can be sold at a price of $4. Give a linear equation in the form p = mn + b that gives the price p they can charge for shirts.

Discover how to create a linear equation to determine shirt pricing based on quantity sold. This problem uses the slope-intercept formula and involves data points such as 9000 shirts sold at $67 and 16000 shirts sold at $4. Ideal for students in Grades 9-11 learning algebra and linear functions.

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