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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.

The points (-5, -3) and (-7, 7) fall on a particular line. What is its equation in point-slope form?

Learn how to find the equation of a line passing through points (-5, -3) and (-7, 7) using point-slope form. This algebraic problem teaches key concepts like calculating slope and applying the point-slope formula, suitable for students in Grades 8-10.

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