Question

find the point-slope equation of the line perpendicular to the line y=2x+5 passing through the point (1,4). (please include solution. thank you very much)

Answer 1

First off, let's keep in mind that perpendicular lines have negative reciprocal slopes, hmmm so what's the slope of y = 2x + 5?

well, notice, that equation is in slope-intercept form, thus   $\bf y=\stackrel{slope}{2}x+5$.

$\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {2\implies \stackrel{slope}{\cfrac{2}{1}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{1}{2}}\qquad \stackrel{negative~reciprocal}{-\cfrac{1}{2}}}$

so, we're really looking for the equation of a line whose slope is -1/2 and runs through 1,4.

$\bf (\stackrel{x_1}{1}~,~\stackrel{y_1}{4})\qquad \qquad \qquad slope = m\implies -\cfrac{1}{2} \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-4=-\cfrac{1}{2}(x-1)$