Question

Theequationoflinepis
y
=
–8
3
x
+ 9
. The
equationoflineqis
y
=
3
8
x
+ 9
. Are
linepandlineqparallelorperpendicular?

Answer 1

To determine whether the two lines \( p \) and \( q \) are parallel or perpendicular, we should first identify their slopes.

1. For line \( p \):
\[
y = -\frac{8}{3}x + 9
\]
The slope of line \( p \) is \( m_p = -\frac{8}{3} \).

2. For line \( q \):
\[
y = \frac{3}{8}x + 9
\]
The slope of line \( q \) is \( m_q = \frac{3}{8} \).

Next, we determine whether the lines are parallel, perpendicular, or neither.

- **Parallel Lines**: Two lines are parallel if their slopes are equal (\( m_p = m_q \)).

- **Perpendicular Lines**: Two lines are perpendicular if the product of their slopes is \( -1 \) (\( m_p \cdot m_q = -1 \)).

Now let's calculate the product of the slopes:
\[
m_p \cdot m_q = \left(-\frac{8}{3}\right) \cdot \left(\frac{3}{8}\right)
\]

Calculating this:
\[
m_p \cdot m_q = -\frac{8 \cdot 3}{3 \cdot 8} = -1.
\]

Since the product of the slopes is \( -1 \), the lines \( p \) and \( q \) are **perpendicular**.