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7. The table below shows the birth rate (per 1000) per year in the United States according to data from the National Center for Health Statistics. Let [tex]\( x \)[/tex] represent the number of years since 2000 with [tex]\( x = 0 \)[/tex] representing the year 2000. Let [tex]\( y \)[/tex] represent the birth rate per 1000 population. Write the slope-intercept form of the equation for the line of fit using the points representing 2001 and 2010. Round to the nearest hundredth.

[tex]\[
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Year & 2010 & 2011 & 2012 & 2013 & 2014 & 2015 & 2016 & 2017 & 2018 & 2019 & 2020 \\
\hline
Birth rate (per 1000) & 15.7 & 15.5 & 15.4 & 15.1 & 15.0 & 15.1 & 15.1 & 15.2 & 15.0 & 14.8 & 15.0 \\
\hline
\end{tabular}
\][/tex]


Sagot :

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]