To determine the slope of line \( q \), which is perpendicular to line \( p \), we need to first identify the slope of line \( p \).
The equation of line \( p \) is given by:
\[ y = \frac{4}{3}x + 8 \]
The slope \( m_p \) of line \( p \) is \( \frac{4}{3} \).
For two lines to be perpendicular, the product of their slopes must equal \(-1\). If the slope of line \( q \) is \( m_q \), we have:
\[ m_p \times m_q = -1 \]
Substituting in the known slope of line \( p \):
\[ \frac{4}{3} \times m_q = -1 \]
To find \( m_q \), we can solve for \( m_q \):
\[ m_q = -\frac{1}{\left(\frac{4}{3}\right)} \]
This can be simplified by multiplying by the reciprocal:
\[ m_q = -\frac{1 \times 3}{4} = -\frac{3}{4} \]
Thus, the slope of line \( q \) is
\[ \boxed{-\frac{3}{4}} \]