To find the slope of the line given the points \((5, 1)\), \((6, 2)\), \((7, 3)\), and \((8, 4)\), we can use the formula for the slope, which is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use any two points from the list. Let's take the points \((5, 1)\) and \((6, 2)\) as an example:
\[ m = \frac{2 - 1}{6 - 5} = \frac{1}{1} = 1 \]
Now, let's verify the slope using other pairs of points to ensure consistency:
Using points \((6, 2)\) and \((7, 3)\):
\[ m = \frac{3 - 2}{7 - 6} = \frac{1}{1} = 1 \]
Using points \((7, 3)\) and \((8, 4)\):
\[ m = \frac{4 - 3}{8 - 7} = \frac{1}{1} = 1 \]
All calculations give us the same slope.
Thus, the slope of the line is \( \boxed{1} \).