To find the equation of the line passing through the points $(-1, -6)$ and $(1, -2)$, we will follow these steps:

Step 1: Calculate the slope (m)

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the given points $(-1, -6)$ and $(1, -2)$: $m = \frac{-2 - (-6)}{1 - (-1)} = \frac{-2 + 6}{1 + 1} = \frac{4}{2} = 2$ Thus, the slope $m = 2$.

Step 2: Use the point-slope form

The equation of a line with slope $m$ passing through a point $(x_1, y_1)$ is given by the point-slope form: $y - y_1 = m(x - x_1)$ Using the point $(-1, -6)$ and the slope $m = 2$: $y - (-6) = 2(x - (-1))$ This simplifies to: $y + 6 = 2(x + 1)$

Step 3: Simplify to slope-intercept form

Now, expand and simplify the equation: $y + 6 = 2(x + 1)$ $y + 6 = 2x + 2$ $y = 2x + 2 - 6$ $y = 2x - 4$

Thus, the equation of the line is: $y = 2x - 4$

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. What would the equation of the line be if it passed through the points $(0, 0)$ and $(1, 2)$?
2. How do we find the y-intercept directly from the point-slope form?
3. How can you verify if another point lies on this line?
4. How do you convert from point-slope form to standard form?
5. What is the significance of the slope in the context of the line's steepness?

Tip: Always check your final equation by substituting the points back into it to ensure both satisfy the equation.