To find the equation of a line that is parallel to a given line and passes through a specific point, we need to follow these steps:
1. Calculate the slope of the given line:
The given points are [tex]\((8,9)\)[/tex] and [tex]\((-12,-7)\)[/tex].
The formula to find the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates:
[tex]\[
\text{slope} = \frac{-7 - 9}{-12 - 8} = \frac{-16}{-20} = \frac{4}{5}
\][/tex]
2. Understand that parallel lines have the same slope:
The line we need to find is parallel to the above line, so it also has a slope of [tex]\(\frac{4}{5}\)[/tex].
3. Use the point-slope form to write the equation of the parallel line:
The point-slope form is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes. Here, the point through which the parallel line passes is [tex]\((-5, -15)\)[/tex].
Substituting the slope [tex]\( \frac{4}{5} \)[/tex] and the point [tex]\((-5, -15)\)[/tex] into the point-slope form:
[tex]\[
y - (-15) = \frac{4}{5}(x - (-5))
\][/tex]
Simplifying this:
[tex]\[
y + 15 = \frac{4}{5}(x + 5)
\][/tex]
So, after deriving the equation in the proper form, the correct choice is:
D. [tex]\(y + 15 = \frac{4}{5}(x + 5)\)[/tex]
