To write the slope-intercept equation for a line passing through the points $(-4, 3)$ and $(2, -9)$, we follow these steps:

Step 1: Find the Slope ($m$)

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Substituting the given points $(-4, 3)$ and $(2, -9)$:

$m = \frac{-9 - 3}{2 - (-4)} = \frac{-12}{6} = -2$

Step 2: Use the Point-Slope Form

The point-slope form of a line's equation is:

$y - y_1 = m(x - x_1)$

We can use either of the given points, but let's use $(-4, 3)$. Substituting the values:

$y - 3 = -2(x + 4)$

Step 3: Convert to Slope-Intercept Form

The slope-intercept form is $y = mx + b$. Distribute and simplify the equation:

$y - 3 = -2x - 8$

Add 3 to both sides:

$y = -2x - 5$

So, the slope-intercept equation of the line is:

$y = -2x - 5$

Do you want more details or have any questions?

Here are some related questions:

1. How do you find the slope when two points are given?
2. What does the slope of a line represent?
3. How do you convert from point-slope form to slope-intercept form?
4. What is the significance of the y-intercept in the slope-intercept equation?
5. How can you graph the equation of a line using the slope and y-intercept?

Tip: Always check your calculations by plugging the points back into the equation to ensure they satisfy the equation.