To determine the slope of a line that is perpendicular to the given line, let's examine the slope of the provided line and then find the slope of the corresponding perpendicular line.
1. Identify the Slope of the Given Line:
The equation of the given line is:
[tex]\[
y = -\frac{1}{4} x - \frac{2}{3}
\][/tex]
In the slope-intercept form of a linear equation, [tex]\(y = mx + b\)[/tex], the coefficient of [tex]\(x\)[/tex] (denoted by [tex]\(m\)[/tex]) represents the slope of the line. Therefore, the slope [tex]\(m\)[/tex] of the given line is:
[tex]\[
m = -\frac{1}{4}
\][/tex]
2. Determine the Slope of the Perpendicular Line:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. To find the negative reciprocal, we follow these steps:
- Take the reciprocal of [tex]\(-\frac{1}{4}\)[/tex]. The reciprocal of [tex]\(-\frac{1}{4}\)[/tex] is [tex]\(-4\)[/tex].
- Change the sign of the reciprocal to its opposite. The opposite sign of [tex]\(-4\)[/tex] is [tex]\(4\)[/tex].
Thus, the slope of the line that is perpendicular to the given line is:
[tex]\[
m_{\perpendicular} = 4
\][/tex]
So, the slope of a line perpendicular to the line [tex]\(y = -\frac{1}{4} x - \frac{2}{3}\)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
