To find the linear equation that relates the price $y$ they can charge for selling $x$ shirts, we need to determine the slope and the y-intercept of the line using the given historical data points.

Given Data Points

• $(x_1, y_1) = (7000, 14)$: 7000 shirts can be sold at a price of $14.
• $(x_2, y_2) = (8000, 9)$: 8000 shirts can be sold at a price of $9.

Step-by-Step Solution

1. Calculate the Slope ($m$):

   The slope $m$ of a line that passes through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$

   Substituting the given points: $m = \frac{9 - 14}{8000 - 7000}$

   Calculate the slope: $m = \frac{-5}{1000} = -0.005$

   So, the slope $m$ is $-0.005$.

2. Determine the Y-Intercept ($b$):

   The equation of a line in slope-intercept form is: $y = mx + b$

   We can use one of the points to solve for $b$. Let's use the point $(7000, 14)$:

   $14 = -0.005(7000) + b$

   Solving for $b$:

   $14 = -35 + b \implies b = 14 + 35 = 49$

   So, the y-intercept $b$ is 49.

3. Formulate the Linear Equation:

   Now, substituting the values of $m$ and $b$ into the slope-intercept form equation: $y = -0.005x + 49$

Final Answer

The linear equation relating the price $y$ they can charge for selling $x$ shirts is:

$y = -0.005x + 49$

Would you like further clarification or have any other questions?

Follow-up Questions

1. What is the interpretation of the slope in this context?
2. How does the y-intercept affect the pricing strategy for the business?
3. Can we use this linear equation to predict prices for any number of shirts, or are there limitations?
4. If the company wants to sell 10,000 shirts, what price should they set according to this model?
5. How many shirts would the company sell if they set the price to $0 according to this model?

Tip: When working with linear equations, remember that the slope represents the rate of change. In business contexts, it can often provide insights into how changes in one variable (like price) might affect another (like sales volume).