To determine 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 slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the coordinates [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex]:
[tex]\[
m = \frac{3 - 7}{1 - (-4)} = \frac{3 - 7}{1 + 4} = \frac{-4}{5} = -\frac{4}{5}
\][/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 slope of the original line. If the slope of the original line is [tex]\(m\)[/tex], the slope of the perpendicular line [tex]\(m_{\perpendicular}\)[/tex] is:
[tex]\[
m_{\perpendicular} = -\frac{1}{m}
\][/tex]
Given the original slope [tex]\(m = -\frac{4}{5}\)[/tex], the slope of the perpendicular line is:
[tex]\[
m_{\perpendicular} = -\frac{1}{-\frac{4}{5}} = \frac{5}{4}
\][/tex]
3. Match the perpendicular slope with the options provided:
We are looking for an equation with a slope of [tex]\(\frac{5}{4}\)[/tex]. Let's examine the slopes in each option:
- Option A: [tex]\(y = \frac{5}{4} x + 8\)[/tex] has a slope of [tex]\(\frac{5}{4}\)[/tex]
- Option B: [tex]\(y = -\frac{4}{5} x + 6\)[/tex] has a slope of [tex]\(-\frac{4}{5}\)[/tex]
- Option C: [tex]\(y = -\frac{5}{4} x - 2\)[/tex] has a slope of [tex]\(-\frac{5}{4}\)[/tex]
- Option D: [tex]\(y = \frac{4}{5} x - 3\)[/tex] has a slope of [tex]\(\frac{4}{5}\)[/tex]
The correct option that has a slope of [tex]\(\frac{5}{4}\)[/tex] is Option A.
Therefore, the equation that represents a line perpendicular to the line passing through [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex] is:
[tex]\[
\boxed{y = \frac{5}{4} x + 8}
\][/tex]
