Question

y = 2/3x - 3
y = -3/2x + 2

Which statement about the lines is true?

Answer options with 4 options
A.
The lines are the same.

B.
The lines are parallel.

C.
The lines are perpendicular.

D.
The lines intersect but are not perpendicular.

Answer 1

To determine the relationship between the two lines given by the equations:

1. \( y = \frac{2}{3}x - 3 \)
2. \( y = -\frac{3}{2}x + 2 \)

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:

1. **Parallel lines** have the same slope. Clearly, \( m_1 \neq m_2 \), so the lines are not parallel.
2. **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.

3. 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.**