Question:

Step 1: Identify the median PS. The median PS connects vertex $P(2, 2)$ to the midpoint $S$ of the opposite side $QR$. Step 2: Find the coordinates of the midpoint S. Given $Q(6, -1)$ and $R(7, 3)$, the coordinates of the midpoint $S$ are: $$S = \left(\frac{6+7}{2}, \frac{-1+3}{2}\right) = \left(\frac{13}{2}, \frac{2}{2}\right) = \left(\frac{13}{2}, 1\right)$$ Step 3: Calculate the slope of the median PS. Given $P(2, 2)$ and $S\left(\frac{13}{2}, 1\right)$, the slope $m_{PS}$ is: $$m_{PS} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 2}{\frac{13}{2} - 2} = \frac{-1}{\frac{13-4}{2}} = \frac{-1}{\frac{9}{2}} = -\frac{2}{9}$$ Step 4: Determine the equation of the line parallel to PS passing through $(1, -1)$. A line parallel to PS has the same slope, so $m = -\frac{2}{9}$. Using the point-slope form $y - y_1 = m(x - x_1)$ with the point $(1, -1)$: $$y - (-1) = -\frac{2}{9}(x - 1)$$ $$y + 1 = -\frac{2}{9}(x - 1)$$ Multiply both sides by 9 to eliminate the fraction: $$9(y + 1) = -2(x - 1)$$ $$9y + 9 = -2x + 2$$ Rearrange the equation into the standard form $Ax + By + C = 0$: $$2x + 9y + 9 - 2 = 0$$ $$2x + 9y + 7 = 0$$