The sales of a small company were ​$46,000 in its third year of operation and ​$88,000 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 ​$110 comma 000.

We are given that the sales of a small company were $46,000 in its third year of operation (i.e., when $ x = 3 $) and $88,000 in its sixth year of operation (i.e., when $ x = 6 $). We are asked to:

1. Find the slope of the sales line and give an equation for the line in the form $ y = mx + b $.
2. Use the equation to find how many years must pass before the sales surpass $110,000.

### Part (a): Find the slope and equation of the line.

We have two points on the line:
- $ (x_1, y_1) = (3, 46,000) $
- $ (x_2, y_2) = (6, 88,000) $

#### Step 1: Find the slope.
The formula for the slope $ m $ of a line through two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the values:
\[
m = \frac{88,000 - 46,000}{6 - 3} = \frac{42,000}{3} = 14,000
\]
So, the slope is $ m = 14,000 $.

#### Step 2: Find the equation of the line.
The general form of a line is:
\[
y = mx + b
\]
We already know $ m = 14,000 $. Now, we need to find $ b $, the y-intercept. To do this, substitute one of the given points (let's use $ (3, 46,000) $) into the equation:

\[
46,000 = 14,000(3) + b
\]
\[
46,000 = 42,000 + b
\]
\[
b = 46,000 - 42,000 = 4,000
\]

Thus, the equation of the sales line is:
\[
y = 14,000x + 4,000
\]

### Part (b): Find how many years must pass before the sales surpass $110,000.

We need to find the value of $ x $ such that $ y = 110,000 $.

Using the equation $ y = 14,000x + 4,000 $, set $ y = 110,000 $ and solve for $ x $:

\[
110,000 = 14,000x + 4,000
\]
\[
110,000 - 4,000 = 14,000x
\]
\[
106,000 = 14,000x
\]
\[
x = \frac{106,000}{14,000} = 7.57
\]

Thus, the sales will surpass $110,000 after approximately 7.57 years. Since we can't have a fraction of a year in this context, we round up to the next whole year, meaning the sales will surpass $110,000 after **8 years**.

### Final Answers:
- (a) The slope of the sales line is 14,000, and the equation is $ y = 14,000x + 4,000 $.
- (b) The sales will surpass $110,000 after 8 years.

Would you like further details or clarifications on any part? 

Here are some related questions to consider:
1. What would the sales be in the 10th year using the same equation?
2. How many years would it take for sales to reach $150,000?
3. What does the slope of the line represent in the context of this problem?
4. How would the equation change if sales in the first year were $20,000?
5. Can we estimate the company’s sales growth if a quadratic function better represented the data?

**Tip:** When interpreting a linear equation, the slope represents the rate of change, while the intercept represents the starting point or initial value.

The sales of a small company were $46,000 in its third year of operation and $88,000 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 = mx + b. (b) Use your answer from part (b) to find out how many years must pass before the sales surpass $110,000.

Learn how to calculate the slope and write the equation of a sales growth line based on data from years 3 and 6 of a small company. Also, determine how many years it will take for the sales to exceed $110,000 using a step-by-step approach. Suitable for Grades 9-10.

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