To determine the slope of a line that is perpendicular to the line given by the equation [tex]\(y = \frac{4}{5}x - 3\)[/tex], we need to follow these steps:
1. Identify the slope of the given line:
The given line is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope. From the equation [tex]\(y = \frac{4}{5}x - 3\)[/tex], we can see that the slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{4}{5}\)[/tex].
2. Understand the relationship between slopes of perpendicular lines:
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. This means if one line has a slope [tex]\(m\)[/tex], the perpendicular line will have a slope of [tex]\(-\frac{1}{m}\)[/tex].
3. Find the negative reciprocal of the given slope:
The slope of the given line is [tex]\(\frac{4}{5}\)[/tex]. To find the slope of the line perpendicular to it, we take the negative reciprocal. The negative reciprocal of [tex]\(\frac{4}{5}\)[/tex] is:
[tex]\[
-\frac{1}{\frac{4}{5}} = -\frac{5}{4}
\][/tex]
4. Simplify the result:
The slope of the line that is perpendicular to the given line [tex]\(y = \frac{4}{5}x - 3\)[/tex] is [tex]\(-\frac{5}{4}\)[/tex].
Thus, the correct answer is [tex]\( \boxed{-\frac{5}{4}} \)[/tex] (option D).
