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 original line:
- 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]
- Insert the values from the given points:
[tex]\[
m = \frac{3 - 7}{1 - (-4)} = \frac{-4}{5} = -0.8
\][/tex]
2. Determine the slope of the perpendicular line:
- The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. If the original slope is [tex]\( m \)[/tex], the perpendicular slope [tex]\( m_{\perp} \)[/tex] is:
[tex]\[
m_{\perp} = -\frac{1}{m}
\][/tex]
- Given the original slope [tex]\( m = -0.8 \)[/tex]:
[tex]\[
m_{\perp} = -\frac{1}{-0.8} = \frac{1}{0.8} = 1.25
\][/tex]
3. Identify the correct equation:
- We know the perpendicular slope is [tex]\( 1.25 \)[/tex] or [tex]\(\frac{5}{4}\)[/tex].
- Now, we look at the options to find the equation with this slope:
- Option A: [tex]\( y = \frac{5}{4} x + 8 \)[/tex]
- Option B: [tex]\( y = -\frac{5}{4} x - 2 \)[/tex]
- Option C: [tex]\( y = -\frac{4}{5} x + 6 \)[/tex]
- Option D: [tex]\( y = \frac{4}{5} x - 3 \)[/tex]
- From these options, the equation with a slope of [tex]\(\frac{5}{4} = 1.25 \)[/tex] is Option A.
So, the equation representing a line that is 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]
