In 2006, the percentage of 18- to 29-year-olds
who used social networking sites was 49%. In 2013, that
percentage had risen to 89%. Use the year as the x-coordinate and the percentage
as the y-coordinate. Find the equation of the line that
contains the data points.

To find the equation of the line that passes through two given data points, we first need to use the formula for the slope of a line, and then find the equation in slope-intercept form \( y = mx + b \).

### Step 1: Identify the data points
The points we are given are:
- In 2006, 49% used social networking sites. This corresponds to the point \( (2006, 49) \).
- In 2013, 89% used social networking sites. This corresponds to the point \( (2013, 89) \).

### Step 2: Find the slope \( m \)
The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the values from the two points:

\[
m = \frac{89 - 49}{2013 - 2006} = \frac{40}{7} \approx 5.71
\]

### Step 3: Use the slope-intercept form of a line \( y = mx + b \)
We now use the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We already know the slope \( m = \frac{40}{7} \).

Now, pick one of the points (we'll use \( (2006, 49) \)) and substitute it into the equation to solve for \( b \):

\[
49 = \frac{40}{7}(2006) + b
\]

Solving for \( b \):

\[
49 = \frac{40}{7} \times 2006 + b
\]

\[
49 = \frac{80240}{7} + b
\]

\[
49 = 11462.86 + b
\]

\[
b = 49 - 11462.86 \approx -11413.86
\]

### Step 4: Final equation
The equation of the line is:

\[
y = \frac{40}{7}x - 11413.86
\]

This is the equation of the line that represents the percentage of 18- to 29-year-olds who used social networking sites over time from 2006 to 2013.

Would you like further details on this or to discuss any specific part of the solution?

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### Related questions:
1. How do you interpret the slope in this context?
2. What would the equation predict for the percentage of users in 2015?
3. Can this model be used to predict the trend beyond 2013? Why or why not?
4. How would the equation change if we used a different set of years?
5. How does linear regression differ from simply connecting two points?

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### Tip:
Whenever calculating a slope or working with data points, remember that the slope gives the rate of change between the variables. In this case, it shows how quickly the percentage of social media use grew per year.

In 2006, the percentage of 18- to 29-year-olds who used social networking sites was 49%. In 2013, that percentage had risen to 89%. Use the year as the x-coordinate and the percentage as the y-coordinate. Find the equation of the line that contains the data points.

Learn how to find the linear equation representing the growth in social media usage among 18- to 29-year-olds from 2006 to 2013. This problem involves calculating the slope and finding the equation in slope-intercept form, making it ideal for high school students in Grades 9-12.

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