Questions: In the xy-plane, points R, S, and T have coordinates (5,6),(5,1), and (8,1), respectively. How much greater is the distance from R to S than the distance from S to T?

To solve this problem, we need to calculate the distances between the given points in the $xy$-plane and then find the difference between these distances.

1. Distance from $R$ to $S$:

   The coordinates of points $R$ and $S$ are $R(5, 6)$ and $S(5, 1)$. Since these points have the same $x$-coordinate, the distance between them is simply the difference in their $y$-coordinates.

   $$\text{Distance from } R \text{ to } S = |6 - 1| = 5$$

2. Distance from $S$ to $T$:

   The coordinates of points $S$ and $T$ are $S(5, 1)$ and $T(8, 1)$. Since these points have the same $y$-coordinate, the distance between them is simply the difference in their $x$-coordinates.

   $$\text{Distance from } S \text{ to } T = |8 - 5| = 3$$

3. Difference in distances:

   Now, we find how much greater the distance from $R$ to $S$ is compared to the distance from $S$ to $T$.

   $$\text{Difference} = 5 - 3 = 2$$

Therefore, the distance from $R$ to $S$ is 2 units greater than the distance from $S$ to $T$.