To find the value of \( k \) given the endpoints \( A(8, k) \) and \( B(7, -3) \) with a slope of 5, we start with the formula for the slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
For our points \( A(8, k) \) and \( B(7, -3) \), we can substitute:
- \( x_1 = 8 \), \( y_1 = k \)
- \( x_2 = 7 \), \( y_2 = -3 \)
Now, plug these values into the slope formula:
\[ 5 = \frac{-3 - k}{7 - 8} \]
Simplifying the denominator:
\[ 5 = \frac{-3 - k}{-1} \]
This can be simplified to:
\[ 5 = 3 + k \]
Now, solving for \( k \):
\[ k = 5 - 3 \] \[ k = 2 \]
Thus, the value of \( k \) is \( \boxed{2} \).