1. **State the problem:** Find the equation of the line perpendicular to the line $y = 4x + 1$ that passes through the point $(8, -3)$. 2. **Recall the slope of the given line:** The given line is in slope-intercept form $y = mx + b$, where $m$ is the slope. Here, $m = 4$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, $$m_{\perp} = -\frac{1}{4}$$ 4. **Use point-slope form:** The equation of a line with slope $m$ passing through point $(x_1, y_1)$ is $$y - y_1 = m(x - x_1)$$ Substitute $m = -\frac{1}{4}$ and point $(8, -3)$: $$y - (-3) = -\frac{1}{4}(x - 8)$$ 5. **Simplify the equation:** $$y + 3 = -\frac{1}{4}x + 2$$ 6. **Isolate $y$ to get slope-intercept form:** $$y = -\frac{1}{4}x + 2 - 3$$ $$y = -\frac{1}{4}x - 1$$ **Final answer:** The equation of the line perpendicular to $y = 4x + 1$ passing through $(8, -3)$ is $$y = -\frac{1}{4}x - 1$$