To find the slope-intercept form of the equation that fits the given data, let's use the points corresponding to the years 2001 and 2010.
First, identify the coordinates:
- In 2001, the birth rate is 14.5. Since 2000 corresponds to [tex]\( x = 0 \)[/tex], 2001 corresponds to [tex]\( x = 1 \)[/tex]. Thus, the point for 2001 is [tex]\((1, 14.5)\)[/tex].
- In 2010, the birth rate is 14.0. Since 2010 is 10 years after 2000, it corresponds to [tex]\( x = 10 \)[/tex]. Thus, the point for 2010 is [tex]\((10, 14.0)\)[/tex].
We have two points: [tex]\((1, 14.5)\)[/tex] and [tex]\((10, 14.0)\)[/tex].
1. Calculate the slope (m):
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given points:
[tex]\[ m = \frac{14.0 - 14.5}{10 - 1} = \frac{-0.5}{9} = -0.05555555555555555 \][/tex]
Rounded to the nearest hundredth, the slope [tex]\( m \)[/tex] is:
[tex]\[ m \approx -0.06 \][/tex]
2. Calculate the y-intercept (b):
The slope-intercept form is [tex]\( y = mx + b \)[/tex]. We can rearrange this formula to solve for [tex]\( b \)[/tex]:
[tex]\[ b = y - mx \][/tex]
Using the point [tex]\((1, 14.5)\)[/tex]:
[tex]\[ b = 14.5 - (-0.06 \times 1) = 14.5 + 0.06 = 14.56 \][/tex]
3. Write the equation in slope-intercept form:
Plug in the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ y = -0.06x + 14.56 \][/tex]
Therefore, the slope-intercept form of the equation for the line of fit using the points representing the years 2001 and 2010 is:
[tex]\[ y = -0.06x + 14.56 \][/tex]
