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To solve this problem, we need to find a linear equation that represents the sales $ S $ of a certain appliance dealer over time. The sales are approximated by a linear function, where:

- Sales were $13,500 in 1982.
- Sales were $60,500 in 1987.
- $ x = 0 $ represents the year 1982.

### Step 1: Understanding the linear equation

A linear function has the form:

\[
S = mx + b
\]

where:
- $ S $ is the sales amount,
- $ m $ is the slope (rate of change of sales per year),
- $ x $ is the number of years since 1982,
- $ b $ is the y-intercept (sales in the year 1982).

### Step 2: Identifying the values

- When $ x = 0 $ (the year 1982), $ S = 13,500 $. So, $ b = 13,500 $.
- In 1987, $ x = 5 $ (since 1987 - 1982 = 5), and $ S = 60,500 $.

### Step 3: Calculating the slope $ m $

The slope $ m $ is calculated as:

\[
m = \frac{\text{Change in sales}}{\text{Change in years}} = \frac{S_2 - S_1}{x_2 - x_1}
\]

Substitute the values:

\[
m = \frac{60,500 - 13,500}{5 - 0} = \frac{47,000}{5} = 9,400
\]

So, the slope $ m = 9,400 $.

### Step 4: Writing the equation

Now we can plug the values of $ m $ and $ b $ into the linear equation:

\[
S = 9,400x + 13,500
\]

### Step 5: Verifying the answer

From the given options:
- **A. $ S = 9,400x + 13,500 $**

This matches our calculated equation.

### Conclusion

The correct answer is **A. $ S = 9,400x + 13,500 $**.

Would you like more details, or do you have any questions?

### Related Questions:
1. How would the equation change if sales were different in 1987?
2. What does the slope $ m = 9,400 $ represent in the context of this problem?
3. How do we interpret the y-intercept $ b = 13,500 $?
4. What would be the sales in 1990 using this linear model?
5. How can this linear model be used to predict future sales?

**Tip:** When solving for a linear function, always identify the initial value (y-intercept) and the rate of change (slope) from the given data.

Assume that the sales of a certain appliance dealer are approximated by a linear function. Suppose that sales were $13,500 in 1982 and $60,500 in 1987. Let x = 0 represent 1982. Find the equation giving yearly sales S.

Explore how to model sales using a linear function, with sales data from 1982 and 1987. Learn to determine the slope and y-intercept to predict sales. Suitable for high school students studying algebra.

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