To find the slope of line \( t \), which is perpendicular to line \( s \), we first need to determine the slope of line \( s \).
The slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In this case, the points are \((6, 7)\) and \((1, 15)\). Assigning the values:
- \( (x_1, y_1) = (6, 7) \)
- \( (x_2, y_2) = (1, 15) \)
Now we can calculate the slope of line \( s \):
\[ m_s = \frac{15 - 7}{1 - 6} = \frac{8}{-5} = -\frac{8}{5} \]
The slope of line \( s \) is \( -\frac{8}{5} \).
Since line \( t \) is perpendicular to line \( s \), we use the fact that the slopes of two perpendicular lines multiply to \(-1\). If \( m_t \) is the slope of line \( t \), then:
\[ m_s \cdot m_t = -1 \]
Substituting the slope of line \( s \):
\[ \left(-\frac{8}{5}\right) \cdot m_t = -1 \]
To find \( m_t \), we solve for \( m_t \):
\[ m_t = \frac{-1}{-\frac{8}{5}} = \frac{5}{8} \]
Thus, the slope of line \( t \) is
\[ \boxed{\frac{5}{8}} \]