To find the slope-intercept form of the equation using data for the years 2010 and 2011:
1. Identify the coordinates:
- For the year 2010, the birth rate is 15.7 per 1000.
- For the year 2011, the birth rate is 15.5 per 1000.
Convert these years into a form where [tex]\( x \)[/tex] represents the number of years since 2010.
- The year 2010 becomes [tex]\( x = 0 \)[/tex].
- The year 2011 becomes [tex]\( x = 1 \)[/tex].
2. Define the coordinates:
- The first point is [tex]\( (0, 15.7) \)[/tex].
- The second point is [tex]\( (1, 15.5) \)[/tex].
3. Calculate the slope [tex]\( m \)[/tex]:
The slope formula is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the values [tex]\( (x_1, y_1) = (0, 15.7) \)[/tex] and [tex]\( (x_2, y_2) = (1, 15.5) \)[/tex]:
[tex]\[
m = \frac{15.5 - 15.7}{1 - 0} = \frac{-0.2}{1} = -0.20
\][/tex]
4. Determine the y-intercept [tex]\( b \)[/tex]:
The y-intercept can be found using the slope-intercept form [tex]\( y = mx + b \)[/tex]. Using the slope [tex]\( m = -0.20 \)[/tex] and the point [tex]\( (0, 15.7) \)[/tex]:
[tex]\[
15.7 = -0.20 \cdot 0 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 15.7
\][/tex]
5. Form the equation:
Using the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex], the slope-intercept form of the line is:
[tex]\[
y = -0.20x + 15.7
\][/tex]
Therefore, the slope-intercept form of the equation for the line of fit using the points representing 2010 and 2011 is:
[tex]\[
y = -0.20x + 15.7
\][/tex]
