Step-by-step Solution:

Step 1: Find the slope of the given line \[\] To find the slope of the given line, we first rewrite the equation in slope-intercept form $y = mx + b$, where $m$ is the slope: $$ x - 2y = 1 $$ $$ -2y = -x + 1 $$ $$ y = \frac{1}{2}x - \frac{1}{2} $$ The slope of the given line is $m = \frac{1}{2}$. \[\] Step 2: Use the property of perpendicular slopes \[\] $$ m_{\text{perpendicular}} = -\frac{1}{\frac{1}{2}} = -2 $$ Step 3: Use the point-slope form to find the equation \[\] Now, we use the point-slope form of the equation of a line, which is: $$ y - y_1 = m(x - x_1) $$ where $m$ is the slope and $(x_1, y_1)$ is a point on the line. We are given the point $(1, 1)$ and the slope $-2$, so: $$ y - 1 = -2(x - 1) $$ Step 4: Simplify the equation $$ y - 1 = -2x + 2 $$ $$ y = -2x + 3 $$