Question

Write an equation for the line that is parallel to the given line and that passes through the given point.

y = 2/3x + 5 (-3, 8)

Answer 1

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

`y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{2}{3}}x+5\qquad \impliedby \begin{array} \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 2/3 and passes through (-3 , 8)

`(\stackrel{x_1}{-3}~,~\stackrel{y_1}{8})\qquad \qquad \stackrel{slope}{m}\implies \cfrac{2}{3} \\\\\\ \begin{array} \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}{8}=\stackrel{m}{\cfrac{2}{3}}(x-\stackrel{x_1}{(-3)}) \\\\\\ y-8=\cfrac{2}{3}(x+3)\implies y-8=\cfrac{2}{3}x+2\implies y=\cfrac{2}{3}x+10`