To find the slope of the line passing through the points given in the table, we will use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can choose any two points from the table. Let's use the points (3, 14) and (4, 20).
Here, \( (x_1, y_1) = (3, 14) \) and \( (x_2, y_2) = (4, 20) \).
Now, substitute these values into the slope formula:
\[ m = \frac{20 - 14}{4 - 3} = \frac{6}{1} = 6 \]
Thus, the slope of the line is
\[ \boxed{6} \]
To ensure correctness, we can also check the slope using another pair of points, such as (4, 20) and (5, 26):
Using points (4, 20) and (5, 26):
\[ m = \frac{26 - 20}{5 - 4} = \frac{6}{1} = 6 \]
And again, using (5, 26) and (6, 32):
\[ m = \frac{32 - 26}{6 - 5} = \frac{6}{1} = 6 \]
Since the slope is consistent with all pairs chosen, we confirm that the slope of the line is
\[ \boxed{6} \]