To write the equation of the line passing through the points $(-2, -2)$ and $(1, 4)$, we follow these steps:

Step 1: Find the slope of the line

The formula for the slope $m$ is:

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

Substitute the coordinates $(-2, -2)$ as $(x_1, y_1)$ and $(1, 4)$ as $(x_2, y_2)$:

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

The slope $m = 2$.

Step 2: Use the point-slope formula

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

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

Substitute $m = 2$ and one of the points, say $(-2, -2)$:

$y - (-2) = 2(x - (-2))$

Simplify:

$y + 2 = 2(x + 2)$

$y + 2 = 2x + 4$

$y = 2x + 2$

Final Answer:

The equation of the line is:

$y = 2x + 2$

Would you like additional steps or verification? Let me know!

Related Questions:

1. How can we verify that both points satisfy this line equation?
2. What would the line look like if it were parallel to this line through another point, say (3, 5)?
3. How would the equation change if the slope were negative?
4. What is the perpendicular slope to the line found here?
5. How do we convert $y = 2x + 2$ to standard form $Ax + By = C$?

Tip:

When solving for the slope, always double-check the signs in the subtraction to avoid mistakes!