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.