The first step in finding the equation of a line that passes through two points, such as $(5, -4)$ and $(-1, 8)$, is to calculate the slope ($m$) using the slope formula:

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

Here, $(x_1, y_1) = (5, -4)$ and $(x_2, y_2) = (-1, 8)$.

Step-by-step calculation of the slope:

1. Subtract the $y$-coordinates: $8 - (-4) = 8 + 4 = 12$

2. Subtract the $x$-coordinates: $-1 - 5 = -6$

3. Now, compute the slope: $m = \frac{12}{-6} = -2$

So, the slope of the line is $m = -2$.

Next Step: Find the equation in slope-intercept form ($y = mx + b$).

To find the $y$-intercept ($b$), substitute one of the points, say $(5, -4)$, and the slope $m = -2$ into the equation $y = mx + b$:

1. Substitute $x = 5$, $y = -4$, and $m = -2$: $-4 = -2(5) + b$

2. Simplify and solve for $b$: $-4 = -10 + b$ $b = -4 + 10 = 6$

Final Equation:

The equation of the line is: $y = -2x + 6$

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Let me know if you would like to go into more detail on any of the steps or explore other related problems. Here are some follow-up questions:

1. What is the significance of the slope in a line's equation?
2. How would the equation change if the slope were positive?
3. Can you verify the equation by substituting the other point $(-1, 8)$?
4. How can we find the intersection point of this line with another line, say $y = x$?
5. How would you calculate the distance between the two points $(5, -4)$ and $(-1, 8)$?

Tip: When finding the slope between two points, always double-check the order of subtraction to avoid sign mistakes.