1. **State the problem:** Find the equation of the line passing through the point $(2, -1)$ and perpendicular to the line given by $2x - y = 4$. 2. **Rewrite the given line in slope-intercept form:** $$2x - y = 4 \implies -y = 4 - 2x \implies y = 2x - 4$$ The slope of this line is $m = 2$. 3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another with slope $m$ is the negative reciprocal: $$m_{\perp} = -\frac{1}{m} = -\frac{1}{2}$$ 4. **Use point-slope form to find the equation of the perpendicular line:** The point-slope form is: $$y - y_1 = m(x - x_1)$$ Substitute $m = -\frac{1}{2}$ and point $(2, -1)$: $$y - (-1) = -\frac{1}{2}(x - 2)$$ $$y + 1 = -\frac{1}{2}x + 1$$ 5. **Simplify the equation:** $$y = -\frac{1}{2}x + 1 - 1$$ $$y = -\frac{1}{2}x$$ **Final answer:** $$y = -\frac{1}{2}x$$