Calculate the Slopes: Lines Through (4,0) to (0,2) and (0,-6) to (4,0)

To determine the slopes of the two lines, we will use the slope formula:

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's start with the first line:

Line I:

The points are $(4, 0)$ and $(0, 2)$. Applying the slope formula:

$m_1 = \frac{2 - 0}{0 - 4} = \frac{2}{-4} = -\frac{1}{2}$

Thus, the slope of Line I is $-\frac{1}{2}$.

Now, let's calculate the slope for the second line:

Line II:

The points are $(0, -6)$ and $(4, 0)$. Applying the slope formula:

$m_2 = \frac{0 - (-6)}{4 - 0} = \frac{6}{4} = \frac{3}{2}$

Therefore, the slope of Line II is $\frac{3}{2}$.

In conclusion, the slopes are as follows:

$\frac{3}{2} = II \text{ , } -\frac{1}{2} = I$.

The solution matches choice $\text{Choice 3}$.

The correct answer to the problem is: $\frac{3}{2} = II, -\frac{1}{2} = I$.