To find the equation of the line that passes through the points $(-4, -5)$ and $(2, -1)$, we can use the slope-intercept form of a line equation:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Step 1: Find the Slope ($m$)

The formula for the slope 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 $(-4, -5)$ and $(2, -1)$:

$m = \frac{-1 - (-5)}{2 - (-4)} = \frac{-1 + 5}{2 + 4} = \frac{4}{6} = \frac{2}{3}$

So, the slope $m$ is $\frac{2}{3}$.

Step 2: Substitute into the Point-Slope Form

Now that we have the slope, we can use the point-slope form of the line equation to find the equation. The point-slope form is:

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

Using the point $(2, -1)$ and $m = \frac{2}{3}$:

$y - (-1) = \frac{2}{3}(x - 2)$

Simplify this equation:

$y + 1 = \frac{2}{3}x - \frac{4}{3}$

Now, subtract 1 from both sides to isolate $y$:

$y = \frac{2}{3}x - \frac{4}{3} - \frac{3}{3}$

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

Final Answer

The equation of the line is:

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

Would you like further details or have any questions?

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1. How do you determine the y-intercept of a line?
2. Can you explain how to convert between different forms of a line equation?
3. What is the significance of the slope in a line equation?
4. How can you verify if a point lies on a given line?
5. How would the equation change if one of the points were different?

Tip: Always double-check your slope calculation, as a small mistake can lead to an incorrect line equation.