Let's solve this step-by-step.

Part 1: Finding the slope of the line

To find the slope $m$ of the line passing through the points $(-5, 6)$ and $(2, -6)$, we use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1) = (-5, 6)$ and $(x_2, y_2) = (2, -6)$.

Substitute the given points: $m = \frac{-6 - 6}{2 - (-5)} = \frac{-12}{2 + 5} = \frac{-12}{7}$

So, the slope is: $m = \frac{-12}{7}$

Answer to Part 1:

The slope of the line is: $m = \frac{-12}{7}$ Thus, the correct choice is A.

Part 2: Writing the equation in point-slope form

The point-slope form of the equation of a line is: $y - y_1 = m(x - x_1)$ Using $m = \frac{-12}{7}$ and one of the points, say $(-5, 6)$, we substitute these values into the formula: $y - 6 = \frac{-12}{7}(x - (-5)) \quad \text{or} \quad y - 6 = \frac{-12}{7}(x + 5)$

Answer to Part 2:

The equation of the line in point-slope form is: $y - 6 = \frac{-12}{7}(x + 5)$ So the correct choice is A.

Part 3: Writing the equation in slope-intercept form

To write the equation in slope-intercept form $y = mx + b$, we start with the point-slope form: $y - 6 = \frac{-12}{7}(x + 5)$

First, expand the equation: $y - 6 = \frac{-12}{7}x - \frac{12}{7} \times 5$ $y - 6 = \frac{-12}{7}x - \frac{60}{7}$

Now, add 6 to both sides to solve for $y$: $y = \frac{-12}{7}x - \frac{60}{7} + 6$ Convert 6 to a fraction with denominator 7: $y = \frac{-12}{7}x - \frac{60}{7} + \frac{42}{7}$ Simplify: $y = \frac{-12}{7}x - \frac{18}{7}$

Answer to Part 3:

The equation of the line in slope-intercept form is: $y = \frac{-12}{7}x - \frac{18}{7}$ Thus, the correct choice is A.

Part 4: Writing the equation in standard form

The standard form of a line is $Ax + By = C$, where $A$, $B$, and $C$ are integers and $A \geq 0$.

We start from the slope-intercept form $y = \frac{-12}{7}x - \frac{18}{7}$. First, eliminate the fractions by multiplying through by 7: $7y = -12x - 18$

Now rearrange to put it in standard form: $12x + 7y = -18$

Answer to Part 4:

The equation of the line in standard form is: $12x + 7y = -18$

Would you like further details or clarifications on any of these steps?

Here are 5 related questions:

1. How can you determine if two lines are parallel based on their slopes?
2. What is the significance of the y-intercept in the slope-intercept form?
3. How do you convert a line equation from standard form to slope-intercept form?
4. What happens to the slope if two points have the same x-coordinate?
5. Can a vertical line be expressed in slope-intercept form? Why or why not?

Tip: The slope of a line indicates how steep it is, with positive slopes rising and negative slopes falling as you move from left to right.