Question

What is an equation of the line that passes through the point (2, -6) and is parallel

Answer 1

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

$x-2y=8\implies -2y=-x+8\implies y=\cfrac{-x+8}{-2} \\\\\\ y=\cfrac{-x}{-2}+\cfrac{8}{-2}\implies y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{2}}x-4 \qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}$

so we're really looking for the equation of a line whose slope is 1/2 and passes through the point (2 , -6)

$(\stackrel{x_1}{2}~,~\stackrel{y_1}{-6})\qquad\qquad \stackrel{slope}{m}\implies \cfrac{1}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-6)}=\stackrel{m}{\cfrac{1}{2}}(x-\stackrel{x_1}{2}) \\\\\\ y+6=\cfrac{1}{2}x-1\implies y=\cfrac{1}{2}x-7$