The equation of line \( s \) is given as:
\[ y = \frac{7}{2}x + \frac{1}{2} \]
In this equation, the slope of line \( s \) is \( \frac{7}{2} \).
Since line \( t \) is perpendicular to line \( s \), the slope of line \( t \) can be found using the property that the slopes of two perpendicular lines multiply to \( -1 \).
Let \( m_s \) be the slope of line \( s \) and \( m_t \) be the slope of line \( t \).
Thus, we have:
\[ m_s \cdot m_t = -1 \]
Substituting the known slope of line \( s \):
\[ \frac{7}{2} \cdot m_t = -1 \]
To find \( m_t \), we can isolate it:
\[ m_t = -\frac{1}{\frac{7}{2}} = -\frac{1 \cdot 2}{7} = -\frac{2}{7} \]
Therefore, the slope of line \( t \) is
\[ \boxed{-\frac{2}{7}}. \]