1. **State the problem:** We have two lines, L and M. Line L passes through points $(4, -1)$ and $(6, 4)$. Line M is perpendicular to L and crosses the y-axis at $(0, 8)$. We need to find where line M intersects the x-axis. 2. **Find the slope of line L:** The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. $$m_L = \frac{4 - (-1)}{6 - 4} = \frac{5}{2}$$ 3. **Find the slope of line M:** Since M is perpendicular to L, its slope $m_M$ is the negative reciprocal of $m_L$. $$m_M = -\frac{1}{m_L} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5}$$ 4. **Write the equation of line M:** Using point-slope form $y = mx + b$, and given $b=8$ (y-intercept), $$y = -\frac{2}{5}x + 8$$ 5. **Find the x-intercept of line M:** At the x-intercept, $y=0$. Substitute and solve for $x$: $$0 = -\frac{2}{5}x + 8$$ $$\frac{2}{5}x = 8$$ $$x = 8 \times \frac{5}{2}$$ $$x = \cancel{8} \times \frac{5}{\cancel{2}} \times 2 = 20$$ 6. **Conclusion:** The x-intercept of line M is at the point $(20, 0)$. **Final answer:** $(20, 0)$