A clothing business finds there is a linear relationship between the number of shirts, xx, it can sell and the price, yy, it can charge per shirt. In particular, historical data shows that 70007000 shirts can be sold at a price of $14$14, while 80008000 shirts can be sold at a price of $9$9.

a) Give a linear equation relating the price yy they can charge for selling xx shirts. Write the slope as a decimal

  

b) Use your equation to predict the number of shirts sold if the price is $39$39.



b) Use your equation to predict the price if the number of shirts is 10001000.

$ 

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To solve this problem, we need to determine the linear equation that relates the number of shirts sold $ x $ and the price $ y $. Given the historical data points, we can use these to find the slope and intercept of the line.

### a) Finding the Linear Equation

1. **Determine the Slope ($m$)**:
   
   The slope of a line in two-dimensional space is given by the change in $ y $ divided by the change in $ x $. Given the two points:
   - Point 1: $ (x_1, y_1) = (7000, 14) $
   - Point 2: $ (x_2, y_2) = (8000, 9) $

   The slope $ m $ is calculated as:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

   Substituting the values:
   \[
   m = \frac{9 - 14}{8000 - 7000} = \frac{-5}{1000} = -0.005
   \]

2. **Determine the Y-Intercept ($b$)**:
   
   The equation of a line can be written as $ y = mx + b $. Using one of the points to solve for $ b $, let's use $ (7000, 14) $:
   \[
   14 = (-0.005)(7000) + b
   \]

   Solving for $ b $:
   \[
   14 = -35 + b \implies b = 49
   \]

3. **Linear Equation**:

   The linear equation relating the price $ y $ to the number of shirts $ x $ is:
   \[
   y = -0.005x + 49
   \]

### b) Predict the Number of Shirts Sold if the Price is $ y = 39 $

To find the number of shirts $ x $ sold at a price $ y = 39 $, substitute $ y = 39 $ into the equation:

\[
39 = -0.005x + 49
\]

Solving for $ x $:

\[
39 - 49 = -0.005x \implies -10 = -0.005x \implies x = \frac{-10}{-0.005} = 2000
\]

Thus, if the price is $39, then the business can expect to sell **2000 shirts**.

### c) Predict the Price if the Number of Shirts is $ x = 1000 $

To find the price $ y $ when the number of shirts $ x = 1000 $, substitute $ x = 1000 $ into the equation:

\[
y = -0.005(1000) + 49
\]

Calculate $ y $:

\[
y = -5 + 49 = 44
\]

Therefore, if the number of shirts is 1000, the price should be **$44**.

---

### Summary

- **Linear equation**: $ y = -0.005x + 49 $
- **Number of shirts sold if price is $39**: 2000 shirts
- **Price if number of shirts is 1000**: $44

Would you like more details on any part of this solution?

### Follow-up Questions

1. What would the price be if the company wants to sell 5000 shirts?
2. How many shirts can the company sell if it charges $20 per shirt?
3. If the price is reduced by $1, how many more shirts can the company expect to sell?
4. What is the break-even point where no shirts are sold?
5. If the company wants to increase its price by $10, how would that affect the number of shirts sold?

**Tip:** When interpreting a linear relationship, the slope represents the rate of change between two variables. In this case, it shows how much the price changes for each additional shirt sold.

Explore how to predict the number of shirts sold and the price based on historical data using a linear equation. Solve for the slope, intercept, and predict outcomes. Suitable for Grades 7-9.

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