Question

Find an equation for a line parallel to 4x + 5y + 2 = 0, with an x-intercept of 3

Answer 1

`\bf 4x+5y+2=0\implies 5y=-4x-2\implies y=\cfrac{-4x-2}{5} \\\\\\ y=-\cfrac{4}{5}x-\cfrac{2}{5}\impliedby \begin{array}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, notice, the slope of that equation is just -4/5, and any parallel line to it will have the same exact slope.

so we're really looking for the equation of a line whose slope is -4/5 and runs through x-intercept of 3, namely (3,0).

`\bf (\stackrel{x_1}{3}~,~\stackrel{y_1}{0})~\hspace{10em} slope = m\implies -\cfrac{4}{5} \\\\\\ \begin{array}c \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-0=-\cfrac{4}{5}(x-3)\implies y=-\cfrac{4}{5}x+\cfrac{12}{5}`