To solve this problem, we need to follow a sequence of steps 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].
1. Find the slope of the line passing through the points [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex]:
The formula to find 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]
Plugging in the given points:
[tex]\[
m = \frac{3 - 7}{1 - (-4)} = \frac{3 - 7}{1 + 4} = \frac{-4}{5} = -\frac{4}{5}
\][/tex]
2. Find the slope of a line perpendicular to the given line:
The slope of any line that is perpendicular to another line is the negative reciprocal of the slope of the original line. So, if the slope of the given line is [tex]\(-\frac{4}{5}\)[/tex], the slope [tex]\(m'\)[/tex] of the perpendicular line is:
[tex]\[
m' = -\left(-\frac{5}{4}\right) = \frac{5}{4}
\][/tex]
3. Identify the equation with the perpendicular slope:
Now we need to check the given equations to find which one has the slope [tex]\(\frac{5}{4}\)[/tex]. Rewrite each equation in the slope-intercept form [tex]\(y = mx + b\)[/tex] and identify the slope [tex]\(m\)[/tex]:
A. [tex]\(y = -\frac{5}{4}x - 2\)[/tex] [tex]\(\rightarrow\)[/tex] slope is [tex]\(-\frac{5}{4}\)[/tex]
B. [tex]\(y = \frac{4}{5}x - 3\)[/tex] [tex]\(\rightarrow\)[/tex] slope is [tex]\(\frac{4}{5}\)[/tex]
C. [tex]\(y = \frac{5}{4}x + 8\)[/tex] [tex]\(\rightarrow\)[/tex] slope is [tex]\(\frac{5}{4}\)[/tex]
D. [tex]\(y = -\frac{4}{5}x + 6\)[/tex] [tex]\(\rightarrow\)[/tex] slope is [tex]\(-\frac{4}{5}\)[/tex]
Comparing the slopes, we see that option C has the slope [tex]\(\frac{5}{4}\)[/tex], which matches the perpendicular slope we calculated.
Therefore, the correct equation representing a line that is perpendicular to the line passing through [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex] is:
C. [tex]\(y = \frac{5}{4}x + 8\)[/tex]
