To determine whether the two lines \( p \) and \( q \) are parallel or perpendicular, we should first identify their slopes.
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For line \( p \): \[ y = -\frac{8}{3}x + 9 \] The slope of line \( p \) is \( m_p = -\frac{8}{3} \).
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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.
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Parallel Lines: Two lines are parallel if their slopes are equal (\( m_p = m_q \)).
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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.