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.

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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.