1. **Stating the problem:** We are given the line equation $$y = 7x - 5$$ and a point $$(-5, -5)$$. We need to find: - The equation of the line parallel to $$y = 7x - 5$$ passing through $$(-5, -5)$$. - The equation of the line perpendicular to $$y = 7x - 5$$ passing through $$(-5, -5)$$. 2. **Identify the slope of the given line:** The given line is in slope-intercept form $$y = mx + b$$ where $$m$$ is the slope. Here, $$m = 7$$. 3. **Parallel line slope:** Parallel lines have the same slope. So, the slope of the parallel line is also $$7$$. 4. **Equation of the parallel line:** Use point-slope form: $$y - y_1 = m(x - x_1)$$ with point $$(-5, -5)$$ and slope $$7$$. $$y - (-5) = 7(x - (-5))$$ Simplify: $$y + 5 = 7(x + 5)$$ $$y + 5 = 7x + 35$$ Subtract $$5$$ from both sides: $$y = 7x + 30$$ 5. **Perpendicular line slope:** Perpendicular slopes are negative reciprocals. Slope of given line is $$7$$, so perpendicular slope is $$-rac{1}{7}$$. 6. **Equation of the perpendicular line:** Use point-slope form again with $$m = -rac{1}{7}$$ and point $$(-5, -5)$$: $$y - (-5) = -\frac{1}{7}(x - (-5))$$ Simplify: $$y + 5 = -\frac{1}{7}(x + 5)$$ Multiply out: $$y + 5 = -\frac{1}{7}x - \frac{5}{7}$$ Subtract 5: $$y = -\frac{1}{7}x - \frac{5}{7} - 5$$ Convert 5 to fraction: $$5 = \frac{35}{7}$$ So: $$y = -\frac{1}{7}x - \frac{5}{7} - \frac{35}{7} = -\frac{1}{7}x - \frac{40}{7}$$ **Final answers:** - Equation of parallel line: $$y = 7x + 30$$ - Equation of perpendicular line: $$y = -\frac{1}{7}x - \frac{40}{7}$$