how did you model it by a linear equation 

The problem describes a situation where the value of the harvester decreases by the same amount each year. This consistent, year-over-year reduction is a key indicator that the relationship between the value of the harvester and time is linear. 

### Understanding Linear Relationships

A linear relationship is characterized by a constant rate of change, meaning that for each unit increase in one variable, the other variable changes by a fixed amount. This kind of relationship can be represented mathematically by a linear equation of the form:
\[
v(t) = mt + b
\]
where:
- $ v(t) $ represents the value of the harvester after $ t $ years,
- $ m $ represents the slope, or the rate of change (in this case, the annual depreciation),
- $ t $ represents the time in years,
- $ b $ represents the y-intercept, or the initial value of the harvester when $ t = 0 $.

### Why Linear?

In this problem, the harvester's value decreases by the same amount each year. This constant decrease is indicative of a linear depreciation model, where:
- The slope $ m $ is negative because the value is decreasing over time.
- The y-intercept $ b $ is the initial value of the harvester, which is the value when $ t = 0 $.

### Justification Using the Given Information

1. **Initial Value**: The harvester's value starts at $ \$330{,}000 $. This gives us the point $ (0, 330{,}000) $.
2. **Value After 10 Years**: After 10 years, the value is $ \$80{,}000 $. This gives us another point $ (10, 80{,}000) $.

With these two points, we can calculate the slope $ m $ (rate of depreciation per year) as follows:
\[
m = \frac{\text{change in value}}{\text{change in time}} = \frac{80{,}000 - 330{,}000}{10 - 0} = -25{,}000
\]

### Forming the Linear Equation

Using the slope $ m = -25{,}000 $ and the initial value $ b = 330{,}000 $, the linear equation representing the harvester's value over time is:
\[
v(t) = -25{,}000t + 330{,}000
\]

This equation models how the value of the harvester decreases linearly over time, which is consistent with the problem's description of uniform annual depreciation.

Would you like further clarification or have any other questions? Here are some related topics you might find useful:

1. What are the differences between linear and exponential depreciation?
2. How do you derive the equation of a line from two points?
3. How would the equation change if the depreciation were not constant each year?
4. How can you verify if a relationship is linear or not using a graph?
5. What are other types of depreciation methods in accounting?
6. How does linear depreciation differ from reducing balance depreciation?
7. Can this concept of linear depreciation be applied to other assets?
8. What are the limitations of using a linear model for depreciation?

**Tip:** When solving word problems, identifying keywords like "constant," "uniform," or "same amount each year" can often help you determine the type of mathematical model to use.

Explore how to model linear depreciation using a harvester as an example. Learn the linear equation v(t) = -25,000t + 330,000 and understand how to calculate the rate of depreciation and initial value. Suitable for Grades 9-12.

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