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 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 the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of the two points:
[tex]\[
m = \frac{3 - 7}{1 - (-4)} = \frac{-4}{5}
\][/tex]
So, the slope of the line through [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex] is [tex]\( -\frac{4}{5} \)[/tex].
2. Find the Slope of the Perpendicular Line:
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of [tex]\( -\frac{4}{5} \)[/tex] is:
[tex]\[
\frac{5}{4}
\][/tex]
3. Identify the Correct Equation:
We need to find the equation among the given options that has a slope of [tex]\( \frac{5}{4} \)[/tex].
Let's examine each option:
- Option A: [tex]\( y = \frac{4}{5} x - 3 \)[/tex] \\
Slope [tex]\( = \frac{4}{5} \)[/tex] (not correct)
- Option B: [tex]\( y = -\frac{4}{5} x + 6 \)[/tex] \\
Slope [tex]\( = -\frac{4}{5} \)[/tex] (not correct)
- Option C: [tex]\( y = -\frac{5}{4} x - 2 \)[/tex] \\
Slope [tex]\( = -\frac{5}{4} \)[/tex] (not correct)
- Option D: [tex]\( y = \frac{5}{4} x + 8 \)[/tex] \\
Slope [tex]\( = \frac{5}{4} \)[/tex] (correct)
Thus, 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]
