To find the distance between the two points $(7, 1)$ and $(4, 5)$, we use the distance formula, which is derived from the Pythagorean theorem:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Here, $(x_1, y_1) = (7, 1)$ and $(x_2, y_2) = (4, 5)$.

Substituting these values into the formula:

$d = \sqrt{(4 - 7)^2 + (5 - 1)^2}$

Simplify the expressions inside the square root:

$d = \sqrt{(-3)^2 + (4)^2}$

$d = \sqrt{9 + 16}$

$d = \sqrt{25}$

$d = 5$

Thus, the distance between the points $(7, 1)$ and $(4, 5)$ is $5$ units.

Would you like further details or have any questions? Here are some related questions:

1. How do you find the midpoint between the points $(7, 1)$ and $(4, 5)$?
2. What is the distance between the points $(7, 1)$ and $(7, 5)$?
3. How would the distance change if one point were moved along the x-axis?
4. Can the distance formula be used in three dimensions? How?
5. How is the distance formula related to the Pythagorean theorem?

Tip: When working with the distance formula, it's essential to ensure that you're squaring the differences correctly, as this will always result in a non-negative value under the square root.