To find the slope of the line that passes through the points \((4, 1)\) and \((7, 9)\), we can use the formula for the slope \(m\) given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In this case, we can assign the coordinates as follows:
- \((x_1, y_1) = (4, 1)\)
- \((x_2, y_2) = (7, 9)\)
Plugging in these values into the slope formula gives:
\[ m = \frac{9 - 1}{7 - 4} \]
Calculating the numerator and the denominator:
\[ m = \frac{8}{3} \]
Thus, the slope of the line that passes through the points \((4, 1)\) and \((7, 9)\) is
\[ \boxed{\frac{8}{3}}. \]