To find the point that is $\frac{2}{3}$ of the way from point $A(5, 1)$ to point $B(-1, 7)$, we again 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, $m = 2$ and $n = 1$, because we want the point to be $\frac{2}{3}$ of the way from $A$ to $B$.

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

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

Now, calculate the coordinates:

$x = \frac{-2 + 5}{3} = \frac{3}{3} = 1$

$y = \frac{14 + 1}{3} = \frac{15}{3} = 5$

So, the point $\frac{2}{3}$ of the way from $(5, 1)$ to $(-1, 7)$ is $(1, 5)$.

Would you like further details or have any questions?

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Here are five related questions:

1. How do you determine a point that is halfway between two points?
2. What is the formula for finding a point one-third of the way between two points?
3. How do you calculate the slope of a line between two points?
4. How does the section formula extend to finding points on a line segment in three dimensions?
5. Can you find the point that divides the segment in the ratio 3:2?

Tip: When calculating points on a line segment using the section formula, double-check the ratio to ensure it corresponds correctly to the required fraction of the way.