Let's work through the problem step by step.

Part (a): Writing the Linear Equation

We need to write a linear equation that relates the years since the system was purchased ($x$) to its value ($y$) in dollars.

We know that the general form of a linear equation is:

$y = mx + b$

Where:

• $y$ is the value of the system
• $x$ is the number of years since the system was purchased
• $m$ is the slope of the line (rate of depreciation)
• $b$ is the y-intercept (initial value of the system)

Step 1: Calculate the Slope ($m$)

We are given two points:

• When $x = 2$, $y = 68,000$
• When $x = 4$, $y = 37,000$

The slope $m$ is calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{37,000 - 68,000}{4 - 2} = \frac{-31,000}{2} = -15,500$

So, the slope $m = -15,500$.

Step 2: Use the Point-Slope Form to Find $b$

Now that we have the slope, we can use one of the points to find the y-intercept $b$. Let's use the point $(x_1, y_1) = (2, 68,000)$:

$y = mx + b$

Substituting the values we know:

$68,000 = -15,500(2) + b$

$68,000 = -31,000 + b$

$b = 68,000 + 31,000 = 99,000$

So, the y-intercept $b = 99,000$.

Step 3: Write the Final Equation

Now we can write the linear equation:

$y = -15,500x + 99,000$

Part (b): Finding the Purchase Price

The purchase price is the value of the system when $x = 0$. So, we substitute $x = 0$ into the equation:

$y = -15,500(0) + 99,000 = 99,000$

So, the purchase price of the system was $99,000.

Summary

• (a) The linear equation relating the years since purchase $x$ and the value $y$ is:
  $y = -15,500x + 99,000$

• (b) The purchase price of the system is:
  $\text{\$99,000}$

Would you like more details on this solution or have any questions?

---

1. How is the depreciation rate calculated from given data?
2. What happens to the system's value after 5 years based on the linear model?
3. Can the linear model be applied for predicting value at any given time?
4. What are other methods of depreciation aside from the linear model?
5. How would the equation change if the depreciation rate was different?

Tip: Always check the reasonableness of the slope when interpreting real-world data. It helps ensure your model makes sense for the context.