Question: What is the equation of the line that passes through the point $(2,-7)$ and has a slope of $-4$?
1. **State the problem:** Find the equation of the line passing through the point $(2,-7)$ with slope $-4$.
2. **Formula used:** The point-slope form of a line is $$y - y_1 = m(x - x_1)$$ where $m$ is the slope and $(x_1,y_1)$ is a point on the line.
3. **Substitute values:** Here, $m = -4$, $x_1 = 2$, and $y_1 = -7$. So,
$$y - (-7) = -4(x - 2)$$
which simplifies to
$$y + 7 = -4(x - 2)$$
4. **Distribute slope:**
$$y + 7 = -4x + 8$$
5. **Isolate $y$ to get slope-intercept form:**
$$y = -4x + 8 - 7$$
$$y = -4x + 1$$
6. **Final answer:** The equation of the line is $$y = -4x + 1$$.