Question

What is an equation of the line that passes through the point (3,-3) and is parallel to the line 5, x, minus, 3, y, equals, 35x−3y=3?

Answer 1

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

`5x-3y=3\implies 5x=3y+3\implies 5x-3=3y\implies \cfrac{5x-3}{3}=y \\\\\\ \stackrel{\stackrel{\textit{\small m}}{\downarrow }}{\cfrac{5}{3}}x-1=y\qquad \impliedby \qquad \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 we are really looking for the equation of a line whose slope is 5/3 and it passes through (3 , -3)

`(\stackrel{x_1}{3}~,~\stackrel{y_1}{-3})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{5}{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}{(-3)}=\stackrel{m}{\cfrac{5}{3}}(x-\stackrel{x_1}{3}) \implies y +3 = \cfrac{5}{3} ( x -3) \\\\\\ y+3=\cfrac{5}{3}x-5\implies {\Large \begin{array}{llll} y=\cfrac{5}{3}x-8 \end{array}}`