Part (a): Writing the Linear Equation

We are asked to write a linear equation that relates the years since the system was purchased, $x$, and its value, $y$, in dollars.

The general form of a linear equation is:

$y = mx + b$

Where:

• $m$ is the slope (rate of depreciation),
• $b$ is the y-intercept (the value when $x = 0$, or the initial value of the system).

We are given two points from the table:

• At $x = 2$ (2 years after purchase), $y = 59,500$,
• At $x = 4$ (4 years after purchase), $y = 33,500$.

Step 1: Calculate the slope, $m$

The slope $m$ is the rate of change of the value of the system over time. It can be calculated using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the values from the two points:

$m = \frac{33,500 - 59,500}{4 - 2} = \frac{-26,000}{2} = -13,000$

So, the slope is $m = -13,000$, meaning the system depreciates by $13,000 per year.

Step 2: Find the y-intercept, $b$

We now use the slope-intercept form $y = mx + b$, and substitute one of the points (for example, $(2, 59,500)$) to solve for $b$:

$59,500 = -13,000(2) + b$

Simplify:

$59,500 = -26,000 + b$

Add 26,000 to both sides:

$b = 59,500 + 26,000 = 85,500$

So, the y-intercept is $b = 85,500$, which represents the value of the system at the time of purchase.

Final Equation:

The linear equation relating $x$ and $y$ is:

$y = -13,000x + 85,500$

Part (b): Find the purchase price (when $x = 0$)

The purchase price is the value of the system when $x = 0$. From the equation:

$y = -13,000(0) + 85,500 = 85,500$

So, the purchase price of the system was:

$\boxed{85,500 \text{ dollars}}$

Would you like any further explanation on this process or specific details?

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Here are five related questions to explore:

1. How would the equation change if the system depreciated at a different rate?
2. Can you calculate the value of the system after 5 years?
3. What happens to the value of the system as time goes on?
4. How would you graph this depreciation model?
5. How do you calculate the time when the system's value reaches zero?

Tip: To find the slope in any linear relationship, always divide the difference in values by the difference in time (or any independent variable).