To find the midpoint \( M \) of the line segment joining the points \( A(4, -7) \) and \( B(6, 3) \), you can use the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( A \) and \( B \).
Plugging in the coordinates of points \( A \) and \( B \):
\[ M = \left( \frac{4 + 6}{2}, \frac{-7 + 3}{2} \right) \]
Calculating the x-coordinate:
\[ \frac{4 + 6}{2} = \frac{10}{2} = 5 \]
Calculating the y-coordinate:
\[ \frac{-7 + 3}{2} = \frac{-4}{2} = -2 \]
So, the midpoint \( M \) is:
\[ M = (5, -2) \]
Thus, the midpoint of the line segment joining points \( A \) and \( B \) is \( \boxed{(5, -2)} \).