To find a point that is $\frac{1}{3}$ of the way from point $A(5, 1)$ to point $B(-1, 7)$, we can use the section formula. The section formula for a point $P(x, y)$ dividing the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$ is:

$P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$

Here, $m = 1$ and $n = 2$, because we want the point to be $\frac{1}{3}$ of the way from $A$ to $B$.

Given points $A(5, 1)$ and $B(-1, 7)$:

$P(x, y) = \left( \frac{1(-1) + 2(5)}{1+2}, \frac{1(7) + 2(1)}{1+2} \right)$

Now, calculate the coordinates:

$x = \frac{-1 + 10}{3} = \frac{9}{3} = 3$

$y = \frac{7 + 2}{3} = \frac{9}{3} = 3$

So, the point $\frac{1}{3}$ of the way from $(5, 1)$ to $(-1, 7)$ is $(3, 3)$.

Would you like further details or have any questions?

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Here are five related questions:

1. How do you find the midpoint between two points?
2. What is the formula for the distance between two points?
3. How do you find the ratio in which a line segment is divided by a point?
4. How do you generalize the section formula for three dimensions?
5. How can you verify if three points are collinear?

Tip: When using the section formula, always ensure the correct order of points to get the desired direction for the division.