To determine the slope of a line that is perpendicular to [tex]\( y = -\frac{2}{3}x + 5 \)[/tex], we need to understand the relationship between the slopes of perpendicular lines. The main concept here is:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line.
1. Identify the slope of the given line.
The given line is expressed in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For the equation [tex]\( y = -\frac{2}{3}x + 5 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex].
2. Find the negative reciprocal of the given slope.
To find the negative reciprocal of [tex]\( -\frac{2}{3} \)[/tex]:
- First, calculate the reciprocal of [tex]\( -\frac{2}{3} \)[/tex], which is [tex]\( -\frac{3}{2} \)[/tex].
- Then, take the negative of this reciprocal. The negative of [tex]\( -\frac{3}{2} \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].
3. Determine which option corresponds to [tex]\( \frac{3}{2} \)[/tex].
- A. [tex]\( -\frac{2}{3} \)[/tex] is not correct.
- B. [tex]\( -\frac{3}{2} \)[/tex] is not correct.
- C. [tex]\( \frac{3}{2} \)[/tex] is correct.
- D. [tex]\( \frac{2}{3} \)[/tex] is not correct.
Therefore, the slope of a line perpendicular to the line [tex]\( y = -\frac{2}{3}x + 5 \)[/tex] is [tex]\( \frac{3}{2} \)[/tex]. Hence, the correct option is:
C. [tex]\( \frac{3}{2} \)[/tex]
