Question

Find an equation of a line that is parallel to 5x - 3y = 6 and passes through the point (6, –2).

Answer 1

well, bearing in mind that parallel lines have the same exact slope hmmmm wait a second, what's the slope of 5x - 3y = 6 anyway?

$\bf 5x-3y=6\implies -3y=-5x+6\implies y=\cfrac{-5x+6}{-3} \\\\\\ y=-\cfrac{5x}{-3}+\cfrac{6}{-3}\implies y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{5}{3}} x-2\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}$

so, we're really looking for the equation of a line whose slope is 5/3 and runs through (6,-2)

$\bf (\stackrel{x_1}{6}~,~\stackrel{y_1}{-2})~\hspace{10em} slope = m\implies \cfrac{5}{3} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-2)=\cfrac{5}{3}(x-6) \\\\\\ y+2=\cfrac{5}{3}x-10\implies y=\cfrac{5}{3}x-12$