Let's solve the problem step by step.

We are asked to find the equation of the line passing through the points $(-5, -3)$ and $(-7, 7)$ in point-slope form.

Step 1: Find the slope $m$

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Using the points $(-5, -3)$ and $(-7, 7)$:

$m = \frac{7 - (-3)}{-7 - (-5)} = \frac{7 + 3}{-7 + 5} = \frac{10}{-2} = -5$

So, the slope $m = -5$.

Step 2: Write the equation in point-slope form

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

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

We can use either point, but let’s use the point $(-5, -3)$.

Substitute $m = -5$, $x_1 = -5$, and $y_1 = -3$:

$y - (-3) = -5(x - (-5))$

Simplifying:

$y + 3 = -5(x + 5)$

This is the equation of the line in point-slope form.

Would you like further details or have any questions?

Here are some related questions to extend the discussion:

1. How can we convert this equation into slope-intercept form?
2. What is the general form of the equation of this line?
3. What would the equation look like if we used the other point $(-7, 7)$?
4. How do you find the midpoint of the line segment between two points?
5. How would the slope change if we swapped the points?

Tip: When simplifying slope formulas, be careful with signs, especially when both coordinates are negative.