A clothing business finds there is a linear relationship between the number of shirts,n,it can sell and the price,p, it can charge per shirt. In particular, historical data shows that 7000 shirts can be sold at a price of $65, while 10000 shirts can be sold at a price of $50. Give a linear equation in the form p=mn+b that gives the price p they can charge for n shirts.

To determine the linear equation in the form $ p = mn + b $, we will first identify two key points based on the historical data given:

- When 7000 shirts are sold, the price is $65: $ (n_1, p_1) = (7000, 65) $
- When 10000 shirts are sold, the price is $50: $ (n_2, p_2) = (10000, 50) $

### Step 1: Find the slope $ m $
The slope of the line, $ m $, represents the rate at which the price changes with respect to the number of shirts sold. We can calculate the slope using the formula:
\[
m = \frac{p_2 - p_1}{n_2 - n_1}
\]
Substituting the given values:
\[
m = \frac{50 - 65}{10000 - 7000} = \frac{-15}{3000} = -0.005
\]

### Step 2: Use the slope-intercept form
Now that we have the slope, we can substitute it into the linear equation form $ p = mn + b $. We also know one point, say $ (n_1, p_1) = (7000, 65) $, which can be used to solve for $ b $ (the y-intercept).

Using the point-slope formula:
\[
p_1 = m \cdot n_1 + b
\]
Substitute the known values:
\[
65 = -0.005 \cdot 7000 + b
\]
\[
65 = -35 + b
\]
\[
b = 65 + 35 = 100
\]

### Step 3: Final equation
Now we have both the slope $ m = -0.005 $ and the y-intercept $ b = 100 $. Therefore, the linear equation is:
\[
p = -0.005n + 100
\]

This equation gives the price $ p $ the business can charge when selling $ n $ shirts.

Would you like further details or clarifications? Here are five related questions to consider:

1. How would the price change if they sold 9000 shirts?
2. At what number of shirts will the price drop to $40?
3. How would the graph of this linear equation look?
4. What does the y-intercept $ b = 100 $ represent in this context?
5. How would changes in production costs affect the slope of this equation?

**Tip:** The slope represents how sensitive the price is to the number of shirts sold, so a steeper slope would mean larger price changes for small changes in shirt sales.

A clothing business finds there is a linear relationship between the number of shirts, n, it can sell and the price, p, it can charge per shirt. In particular, historical data shows that 7000 shirts can be sold at a price of $65, while 10000 shirts can be sold at a price of $50. Give a linear equation in the form p = mn + b that gives the price p they can charge for n shirts.

Learn how to derive a linear equation for the price of shirts based on the number of shirts sold. Using historical data showing 7000 shirts sold at $65 and 10000 shirts at $50, we calculate the slope and form the equation p = -0.005n + 100. This algebraic solution is perfect for students in Grades 8-10.

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