To determine the relationship between the two lines given by the equations:
- \( y = \frac{2}{3}x - 3 \)
- \( y = -\frac{3}{2}x + 2 \)
we need to analyze their slopes.
The slopes are:
- For the first line \( y = \frac{2}{3}x - 3 \), the slope \( m_1 = \frac{2}{3} \).
- For the second line \( y = -\frac{3}{2}x + 2 \), the slope \( m_2 = -\frac{3}{2} \).
Now, we check the relationship between the slopes:
-
Parallel lines have the same slope. Clearly, \( m_1 \neq m_2 \), so the lines are not parallel.
-
Perpendicular lines have slopes that are negative reciprocals of each other. To check whether the product of the slopes equals \(-1\):
\[ m_1 \cdot m_2 = \left(\frac{2}{3}\right) \cdot \left(-\frac{3}{2}\right) = -1 \]
Since this product equals \(-1\), the lines are perpendicular.
- Lastly, since the lines have different slopes, they are not the same and they will intersect.
Thus, the correct answer is:
C. The lines are perpendicular.