To determine which equation represents a line that is perpendicular to the given equation [tex]\( y = -\frac{2}{3} x - 4 \)[/tex], we need to understand the relationship between the slopes of perpendicular lines.
### Step-by-Step Solution:
1. Identify the Slope of the Given Line:
The equation of the line is provided in the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. For the equation [tex]\( y = -\frac{2}{3} x - 4 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex].
2. Determine the Slope of the Perpendicular Line:
The slopes of two perpendicular lines multiply to -1. In other words, if [tex]\( m_1 \)[/tex] is the slope of one line, and [tex]\( m_2 \)[/tex] is the slope of the line perpendicular to it, then [tex]\( m_1 \times m_2 = -1 \)[/tex].
Given [tex]\( m_1 = -\frac{2}{3} \)[/tex], we need to find [tex]\( m_2 \)[/tex] such that:
[tex]\[
-\frac{2}{3} \times m_2 = -1
\][/tex]
Solving for [tex]\( m_2 \)[/tex]:
[tex]\[
m_2 = \frac{3}{2}
\][/tex]
3. Find the Correct Equation:
The slope of the line perpendicular to the given line is [tex]\( \frac{3}{2} \)[/tex]. Now, looking at the provided options:
A. [tex]\( y = -\frac{2}{3} x + 1 \)[/tex] (Slope: [tex]\(-\frac{2}{3}\)[/tex])
B. [tex]\( y = \frac{2}{3} x + 1 \)[/tex] (Slope: [tex]\(\frac{2}{3}\)[/tex])
C. [tex]\( y = -\frac{3}{2} x + 1 \)[/tex] (Slope: [tex]\(-\frac{3}{2}\)[/tex])
D. [tex]\( y = \frac{3}{2} x + 1 \)[/tex] (Slope: [tex]\(\frac{3}{2}\)[/tex])
The correct choice has to have a slope of [tex]\( \frac{3}{2} \)[/tex].
Therefore, the correct answer is:
D. [tex]\( y = \frac{3}{2} x + 1 \)[/tex]
