Let's start by determining the slope of the line that passes 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 given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of the given points:
[tex]\[
m = \frac{3 - 7}{1 - (-4)} = \frac{3 - 7}{1 + 4} = \frac{-4}{5}
\][/tex]
So, the slope of the line passing through [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex] is [tex]\(\frac{-4}{5}\)[/tex].
Now, we want to find a line that is perpendicular to this line. The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line.
The negative reciprocal of [tex]\(\frac{-4}{5}\)[/tex] is:
[tex]\[
-\frac{1}{\frac{-4}{5}} = 5/4
\][/tex]
Thus, the slope of the line that is perpendicular to the given line is [tex]\(\frac{5}{4}\)[/tex].
Next, let's check the slopes of the given equations to determine which one has a slope of [tex]\(\frac{5}{4}\)[/tex]:
- Equation A: [tex]\(y = \frac{5}{4} x + 8\)[/tex] has a slope of [tex]\(\frac{5}{4}\)[/tex].
- Equation B: [tex]\(y = -\frac{5}{4} x - 2\)[/tex] has a slope of [tex]\(-\frac{5}{4}\)[/tex].
- Equation C: [tex]\(y = -\frac{4}{5} x + 6\)[/tex] has a slope of [tex]\(-\frac{4}{5}\)[/tex].
- Equation D: [tex]\(y = \frac{4}{5} x - 3\)[/tex] has a slope of [tex]\(\frac{4}{5}\)[/tex].
From the above analysis, it is clear that Equation A has a slope of [tex]\(\frac{5}{4}\)[/tex], which matches the slope we found for the perpendicular line.
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]
which corresponds to choice A (Answer 1).
