Let's go through the problem step by step and solve for each pair of points. The task involves finding the slope, vertical intercept, and the equation of the line that passes through each pair of points.

Slope Formula:

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as:

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

Vertical Intercept:

To find the vertical intercept, use the equation of a line $y = mx + b$ and substitute one of the points along with the slope to solve for $b$.

Let's solve them one by one:

1. Points: $(-4, -24)$ and $(3, 18)$:

   • Slope: $m = \frac{18 - (-24)}{3 - (-4)} = \frac{18 + 24}{3 + 4} = \frac{42}{7} = 6$
   • Using point $(-4, -24)$, the equation becomes: $y = 6x + b$ Plug in $(-4, -24)$: $-24 = 6(-4) + b \Rightarrow -24 = -24 + b \Rightarrow b = 0$
   • Equation of the line: $y = 6x$.

2. Points: $(-8, 15)$ and $(12, -45)$:

   • Slope: $m = \frac{-45 - 15}{12 - (-8)} = \frac{-60}{12 + 8} = \frac{-60}{20} = -3$
   • Using point $(-8, 15)$, the equation becomes: $y = -3x + b$ Plug in $(-8, 15)$: $15 = -3(-8) + b \Rightarrow 15 = 24 + b \Rightarrow b = 15 - 24 = -9$
   • Equation of the line: $y = -3x - 9$.

3. Points: $(-12, -47)$ and $(10, 41)$:

   • Slope: $m = \frac{41 - (-47)}{10 - (-12)} = \frac{41 + 47}{10 + 12} = \frac{88}{22} = 4$
   • Using point $(-12, -47)$, the equation becomes: $y = 4x + b$ Plug in $(-12, -47)$: $-47 = 4(-12) + b \Rightarrow -47 = -48 + b \Rightarrow b = -47 + 48 = 1$
   • Equation of the line: $y = 4x + 1$.

4. Points: $(9, 6)$ and $(9, 11)$:

   • Slope: $m = \frac{11 - 6}{9 - 9} = \frac{5}{0}$ Since the denominator is 0, the slope is undefined, so the slope is DNE (Does Not Exist), meaning the line is vertical.

5. Points: $(-10, -1)$ and $(6, -1)$:

   • Slope: $m = \frac{-1 - (-1)}{6 - (-10)} = \frac{0}{6 + 10} = 0$
   • A slope of 0 means the line is horizontal. The equation of the line is $y = -1$.

Final Results:

1. $(-4, -24)$ and $(3, 18)$:

   • Slope: $6$
   • Vertical Intercept: $0$
   • Equation: $y = 6x$

2. $(-8, 15)$ and $(12, -45)$:

   • Slope: $-3$
   • Vertical Intercept: $-9$
   • Equation: $y = -3x - 9$

3. $(-12, -47)$ and $(10, 41)$:

   • Slope: $4$
   • Vertical Intercept: $1$
   • Equation: $y = 4x + 1$

4. $(9, 6)$ and $(9, 11)$:

   • Slope: Undefined (Vertical line)
   • Vertical Intercept: DNE
   • Equation: Vertical line at $x = 9$

5. $(-10, -1)$ and $(6, -1)$:

   • Slope: $0$
   • Vertical Intercept: $-1$
   • Equation: $y = -1$

Would you like more details on any of the steps or clarifications?

Here are 5 related questions to expand on the concept:

1. How do we derive the equation of a line if the slope is undefined?
2. What are the characteristics of a line with zero slope?
3. How can you verify if a pair of points lies on a given line?
4. How do horizontal and vertical lines differ in terms of their equations?
5. What is the effect of changing the slope of a line on its steepness?

Tip: For vertical and horizontal lines, their equations are quite simple: $x = c$ for vertical lines and $y = c$ for horizontal ones, where $c$ is a constant.