Perpendicular Line 24B3Ff

1. **State the problem:** We are given a line with slope $X = -1$ and a point $(-2, 2)$. We need to find the equation of the line perpendicular to this given line that passes through the point $(-2, 2)$. 2. **Recall the rule for perpendicular slopes:** If a line has slope $m$, then a line perpendicular to it has slope $m_{\perp} = -\frac{1}{m}$. 3. **Calculate the perpendicular slope:** Given $m = -1$, the perpendicular slope is $$m_{\perp} = -\frac{1}{-1} = 1.$$ 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 $m = 1$ and point $(-2, 2)$: $$y - 2 = 1(x - (-2))$$ $$y - 2 = 1(x + 2)$$ 5. **Simplify the equation:** $$y - 2 = x + 2$$ $$y = x + 2 + 2$$ $$y = x + 4.$$ **Final answer:** The line perpendicular to the given line with slope $-1$ passing through $(-2, 2)$ is $$y = x + 4.$$