The problem is to find the slope $m$ of the line passing through the points $P(1,4)$ and $Q(9,8)$. Then, it appears that the calculation of the slope of the line passing through the point $Q(9,8)$ and $V(2,8)$ is desired. The point $U(-6,1)$ seems to be irrelevant to the presented calculation.

1. Problem Description

The problem is to find the slope $m$ of the line passing through the points $P(1,4)$ and $Q(9,8)$. Then, it appears that the calculation of the slope of the line passing through the point $Q(9,8)$ and $V(2,8)$ is desired. The point $U(-6,1)$ seems to be irrelevant to the presented calculation.

2. Solution Steps

First, we will calculate the slope $m_{PQ}$ of the line passing through points $P(1,4)$ and $Q(9,8)$ using the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the coordinates of $P(1,4)$ and $Q(9,8)$, where $x_1 = 1$, $y_1 = 4$, $x_2 = 9$, and $y_2 = 8$, we have:

$m_{PQ} = \frac{8 - 4}{9 - 1}$

$m_{PQ} = \frac{4}{8}$

$m_{PQ} = \frac{1}{2}$

Then we will compute the slope of the line passing through the points $Q(9,8)$ and $V(2,8)$:

$m_{QV} = \frac{8 - 8}{2 - 9} = \frac{0}{-7} = 0$

Note: The image shows $m=\frac{8-1}{9-1} = \frac{7}{8}$. It used the y-coordinate of Q, but the x-coordinate of P. This is incorrect. It also uses the y coordinate of V and x coordinate of U when they are unrelated to each other.

3. Final Answer

The slope of the line passing through $P(1,4)$ and $Q(9,8)$ is $m_{PQ} = \frac{1}{2}$.

The slope of the line passing through $Q(9,8)$ and $V(2,8)$ is $m_{QV} = 0$.

The value computed in the image is $\frac{7}{8}$, but this is the slope of the line between (1,1) and (9,8).