To find the slope of the line that passes through the points \((x_1, y_1) = (6, 5)\) and \((x_2, y_2) = (2, 14)\), we can use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the points into the formula:
\[ m = \frac{14 - 5}{2 - 6} \]
Calculating the differences in the numerator and denominator:
\[ m = \frac{9}{-4} \]
Simplifying this fraction gives:
\[ m = -\frac{9}{4} \]
Thus, the slope of the line that passes through the points \((6, 5)\) and \((2, 14)\) is
\[ \boxed{-\frac{9}{4}}. \]