1. **State the problem:** We are given the line $y = -2x + 8$ and a point $(7, -3)$. We need to find: - The equation of the line perpendicular to $y = -2x + 8$ passing through $(7, -3)$. - The equation of the line parallel to $y = -2x + 8$ passing through $(7, -3)$. 2. **Recall the slope rules:** - The slope of the original line is $m = -2$. - The slope of a line perpendicular to it is the negative reciprocal: $m_\perp = \frac{1}{2}$. - The slope of a line parallel to it is the same: $m_\parallel = -2$. 3. **Use point-slope form:** The point-slope form of a line is: $$y - y_1 = m(x - x_1)$$ where $(x_1, y_1)$ is the point the line passes through. 4. **Find the perpendicular line equation:** Substitute $m = \frac{1}{2}$ and point $(7, -3)$: $$y - (-3) = \frac{1}{2}(x - 7)$$ Simplify: $$y + 3 = \frac{1}{2}x - \frac{7}{2}$$ Subtract 3 from both sides: $$y = \frac{1}{2}x - \frac{7}{2} - 3$$ Express 3 as $\frac{6}{2}$ to combine: $$y = \frac{1}{2}x - \frac{7}{2} - \frac{6}{2} = \frac{1}{2}x - \frac{13}{2}$$ 5. **Find the parallel line equation:** Substitute $m = -2$ and point $(7, -3)$: $$y - (-3) = -2(x - 7)$$ Simplify: $$y + 3 = -2x + 14$$ Subtract 3 from both sides: $$y = -2x + 14 - 3 = -2x + 11$$ **Final answers:** - Perpendicular line: $$y = \frac{1}{2}x - \frac{13}{2}$$ - Parallel line: $$y = -2x + 11$$