To find which equation represents a line that is perpendicular to the line passing through the points [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex], we need to follow these steps:
1. Calculate the slope of the line passing through the given points:
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]
For the points [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex]:
[tex]\[
m = \frac{3 - 7}{1 - (-4)} = \frac{-4}{5} = -\frac{4}{5}
\][/tex]
2. Determine the slope of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of the original line is [tex]\(-\frac{4}{5}\)[/tex], the perpendicular slope [tex]\(m_\perp\)[/tex] is:
[tex]\[
m_\perp = -\left(\frac{1}{-\frac{4}{5}}\right) = \frac{5}{4}
\][/tex]
3. Identify the correct equation:
Now, we need to find the equation from the given choices that has this perpendicular slope. Let's examine the options:
A. [tex]\(y = -\frac{5}{4} x - 2\)[/tex]
This line has a slope of [tex]\(-\frac{5}{4}\)[/tex].
B. [tex]\(y = -\frac{4}{5} x + 6\)[/tex]
This line has a slope of [tex]\(-\frac{4}{5}\)[/tex].
C. [tex]\(y = \frac{4}{5} x - 3\)[/tex]
This line has a slope of [tex]\(\frac{4}{5}\)[/tex].
D. [tex]\(y = \frac{5}{4} x + 8\)[/tex]
This line has a slope of [tex]\(\frac{5}{4}\)[/tex].
The correct equation of the line that is perpendicular to the line passing through [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex] is therefore:
[tex]\[
\boxed{y = \frac{5}{4} x + 8}
\][/tex]
This confirms that option D is the correct answer.
