To find the equation of the line passing through the points $(5, -2)$ and $(8, -9)$, we'll follow these steps:

1. Calculate the slope (m) of the line:

The slope formula is:

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

Substituting the given points $(x_1, y_1) = (5, -2)$ and $(x_2, y_2) = (8, -9)$:

$m = \frac{-9 - (-2)}{8 - 5} = \frac{-9 + 2}{3} = \frac{-7}{3}$

So, the slope $m = \frac{-7}{3}$.

2. Use the point-slope form of the equation:

The point-slope form of a line is:

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

Using the point $(5, -2)$ and the slope $m = \frac{-7}{3}$:

$y - (-2) = \frac{-7}{3}(x - 5)$

This simplifies to:

$y + 2 = \frac{-7}{3}(x - 5)$

3. Final simplified point-slope form:

The equation of the line in fully simplified point-slope form is:

$y + 2 = \frac{-7}{3}(x - 5)$

Would you like more details on any part of the process?

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Here are 5 related questions to deepen your understanding:

1. How do you convert point-slope form to slope-intercept form?
2. What is the significance of the slope in a linear equation?
3. How can you tell if two points create a horizontal or vertical line?
4. How would the equation change if the line passed through $(5, -2)$ and $(8, 9)$?
5. What is the geometric interpretation of a negative slope?

Tip: Always recheck your slope calculation—it’s easy to switch signs!