To determine the slope of the line defined by the points \((0,0)\), \((1,2)\), \((2,4)\), \((3,6)\), and \((4,8)\), we can use the coordinates of any two points to calculate the slope.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's use the points \((0,0)\) and \((1,2)\):
- \(x_1 = 0\), \(y_1 = 0\)
- \(x_2 = 1\), \(y_2 = 2\)
Substituting these values into the slopes formula:
\[ m = \frac{2 - 0}{1 - 0} = \frac{2}{1} = 2 \]
Now, to further verify, we can use another pair of points, for example, \((2,4)\) and \((4,8)\):
- \(x_1 = 2\), \(y_1 = 4\)
- \(x_2 = 4\), \(y_2 = 8\)
Substituting these values gives:
\[ m = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2 \]
The slope is consistent regardless of which points we choose. Therefore, the slope of the line defined by the given points is:
\[ \boxed{2} \]