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 $$m=\frac{y_2 - y_1}{x_2 - x_1}$$. Calculate: $$m_L = \frac{4 - (-1)}{6 - 4} = \frac{5}{2}$$. 3. **Find the slope of line M:** Since M is perpendicular to L, its slope $$m_M$$ satisfies: $$m_M = -\frac{1}{m_L} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5}$$. 4. **Write the equation of line M:** It passes through (0, 8), so its equation in slope-intercept form is: $$y = m_M x + b$$ Since it crosses the y-axis at (0, 8), $$b = 8$$. Thus: $$y = -\frac{2}{5}x + 8$$. 5. **Find the x-intercept of line M:** At the x-intercept, $$y=0$$. Set $$y=0$$ and solve for $$x$$: $$0 = -\frac{2}{5}x + 8$$ Add $$\frac{2}{5}x$$ to both sides: $$\frac{2}{5}x = 8$$ Multiply both sides by $$\cancel{\frac{5}{2}} \times \frac{5}{2}$$ to isolate $$x$$: $$x = 8 \times \frac{5}{2} = 20$$. 6. **Final answer:** The coordinates where line M intersects the x-axis are: $$(20, 0)$$.