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 points [tex]\((-4, 7)\)[/tex] and [tex]\( (1, 3) \)[/tex].
The formula for 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] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Applying the given points:
[tex]\[
m = \frac{3 - 7}{1 - (-4)} = \frac{-4}{1 + 4} = \frac{-4}{5}
\][/tex]
So, the slope of the line passing through these points is [tex]\( -\frac{4}{5} \)[/tex].
2. Determine the slope of the line that is perpendicular.
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. For a slope [tex]\(m\)[/tex], the slope of the perpendicular line is [tex]\(-\frac{1}{m}\)[/tex].
Given the original slope is [tex]\(-\frac{4}{5}\)[/tex], the slope of the perpendicular line is:
[tex]\[
-\frac{1}{\left(-\frac{4}{5}\right)} = \frac{5}{4}
\][/tex]
3. Match the calculated perpendicular slope with the slopes given in the options.
Let's analyze the provided options:
- Option A: [tex]\( y = \frac{4}{5}x - 3 \)[/tex] has a slope of [tex]\( \frac{4}{5} \)[/tex]
- Option B: [tex]\( y = -\frac{4}{5}x + 6 \)[/tex] has a slope of [tex]\(-\frac{4}{5} \)[/tex]
- Option C: [tex]\( y = \frac{5}{4}x + 8 \)[/tex] has a slope of [tex]\( \frac{5}{4} \)[/tex]
- Option D: [tex]\( y = -\frac{5}{4}x - 2 \)[/tex] has a slope of [tex]\(-\frac{5}{4} \)[/tex]
Out of these options, only Option C has the slope [tex]\( \frac{5}{4} \)[/tex], which matches the slope of the perpendicular line we calculated.
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]
