To find the slope of the line that passes through the points \((10, 1)\) and \((1, 6)\), we can use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we can assign:
- \((x_1, y_1) = (10, 1)\)
- \((x_2, y_2) = (1, 6)\)
Now, substitute the values into the formula:
\[ m = \frac{6 - 1}{1 - 10} \]
Calculating the differences:
\[ m = \frac{5}{1 - 10} = \frac{5}{-9} \]
Simplifying the fraction gives:
\[ m = -\frac{5}{9} \]
Thus, the slope of the line that passes through the points \((10, 1)\) and \((1, 6)\) is
\[ \boxed{-\frac{5}{9}}. \]