The tasks in the image involve calculating the slope of a line passing through two given points, writing the equation of the line in point-slope form using the second point, and converting that equation into slope-intercept form.

a) Calculation of the Slope

Given two points, $(-4, 3)$ and $(2, -6)$, the formula for the slope $m$ between these points is:

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

Plugging in the coordinates of the points:

$m = \frac{-6 - 3}{2 + 4} = \frac{-9}{6} = -\frac{3}{2}$

b) Point-Slope Form

Using the second point $(2, -6)$ and the slope $-\frac{3}{2}$, the point-slope form of the line is given by:

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

Substituting the slope and the second point into the equation:

$y - (-6) = -\frac{3}{2}(x - 2)$

$y + 6 = -\frac{3}{2}x + 3$

c) Slope-Intercept Form

To convert the point-slope form to slope-intercept form $y = mx + b$, solve for $y$:

$y = -\frac{3}{2}x + 3 - 6$

$y = -\frac{3}{2}x - 3$

This is the equation of the line in slope-intercept form, where the slope $m = -\frac{3}{2}$ and the y-intercept $b = -3$.

Would you like further details on any of these calculations, or do you have any additional questions? Here are some related queries:

1. How would you find the x-intercept of this line?
2. What happens if you use the first point in the point-slope form?
3. How can you verify the accuracy of the slope-intercept form?
4. Could you calculate the distance between the two points?
5. What would be the midpoint of the segment joining these two points?

Tip: To visually confirm the accuracy of the slope and y-intercept, you could plot the equation on a graph and ensure it passes through the given points.