1. **State the problem:** Find the equation of the line passing through the point $(-1, -7)$ and satisfying the equation $3x + 12y = -6$. 2. **Rewrite the given line equation in slope-intercept form:** Start with the equation: $$3x + 12y = -6$$ Isolate $y$: $$12y = -3x - 6$$ Divide both sides by 12: $$y = \frac{\cancel{12}y}{\cancel{12}} = \frac{-3x - 6}{12} = -\frac{3}{12}x - \frac{6}{12}$$ Simplify the fractions: $$y = -\frac{1}{4}x - \frac{1}{2}$$ So the slope $m$ of the line is $-\frac{1}{4}$. 3. **Check if the point $(-1, -7)$ lies on the line:** Substitute $x = -1$ into the line equation: $$y = -\frac{1}{4}(-1) - \frac{1}{2} = \frac{1}{4} - \frac{1}{2} = -\frac{1}{4}$$ But the $y$-coordinate of the point is $-7$, which is not equal to $-\frac{1}{4}$. So the point does not lie on the line. 4. **Find the equation of the line passing through $(-1, -7)$ with the same slope $m = -\frac{1}{4}$:** Use point-slope form: $$y - y_1 = m(x - x_1)$$ Substitute $m = -\frac{1}{4}$, $x_1 = -1$, $y_1 = -7$: $$y - (-7) = -\frac{1}{4}(x - (-1))$$ Simplify: $$y + 7 = -\frac{1}{4}(x + 1)$$ Distribute: $$y + 7 = -\frac{1}{4}x - \frac{1}{4}$$ Subtract 7 from both sides: $$y = -\frac{1}{4}x - \frac{1}{4} - 7$$ Simplify: $$y = -\frac{1}{4}x - \frac{29}{4}$$ **Final answer:** The equation of the line passing through $(-1, -7)$ with slope $-\frac{1}{4}$ is $$y = -\frac{1}{4}x - \frac{29}{4}$$