The sales of a small company were ​$45 comma 000 in its fourth year of operation and ​$65 comma 000 in its sixth year. Let y represent sales in the xth year of operation. Assume that the data can be approximated by a straight line. 
​(a) Find the slope of the sales​ line, and give an equation for the line in the form y equals mx plus b.
​(b) Use your answer from part​ (b) to find out how many years must pass before the sales surpass ​$120 comma 000

To solve this problem, let's break it down into two parts.

### Part (a): Finding the slope of the sales line and the equation of the line

1. **Identify the points:**  
   We have two points on the line based on the information given:
   - In the fourth year: $(x_1, y_1) = (4, 45000)$
   - In the sixth year: $(x_2, y_2) = (6, 65000)$

2. **Calculate the slope $m$:**  
   The slope of a line is calculated using the formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting the values:
   \[
   m = \frac{65000 - 45000}{6 - 4} = \frac{20000}{2} = 10000
   \]
   So, the slope $m$ is $10000$.

3. **Find the equation of the line:**  
   The general form of the equation of a line is:
   \[
   y = mx + b
   \]
   We can use one of the points to solve for $b$ (the y-intercept). Using the point $(4, 45000)$ and substituting $m = 10000$:
   \[
   45000 = 10000(4) + b
   \]
   \[
   45000 = 40000 + b
   \]
   \[
   b = 45000 - 40000 = 5000
   \]
   Therefore, the equation of the sales line is:
   \[
   y = 10000x + 5000
   \]

### Part (b): Finding the number of years until sales surpass $120,000

We need to find $x$ when $y = 120000$. Using the equation we derived:

\[
120000 = 10000x + 5000
\]

Subtract 5000 from both sides:

\[
115000 = 10000x
\]

Divide both sides by 10000:

\[
x = \frac{115000}{10000} = 11.5
\]

So, the sales will surpass $120,000 after approximately 11.5 years. Since we're dealing with years, we round up to the next whole number, so it will take **12 years** for the sales to surpass $120,000.

### Summary:

- The equation of the sales line is $ y = 10000x + 5000 $.
- The sales will surpass $120,000 in the 12th year of operation.

Do you want more details or have any questions? Here are some related questions:

1. How would the equation change if the sales in the sixth year were $75,000?
2. What would be the sales in the 10th year according to this model?
3. How would you interpret the y-intercept in this context?
4. What does the slope of the line represent in terms of the company's sales?
5. How would you graph this line on a coordinate plane?

**Tip:** Always check if the data can reasonably be modeled by a linear equation before applying linear models.

Learn how to solve a sales growth problem using linear equations. Calculate the slope and y-intercept of the sales line with given sales data ($45,000 in year 4 and $65,000 in year 6). Discover when sales will surpass $120,000 based on this model. Suitable for high school students.

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