Question

A design engineer is mapping out a new neighborhood with parallel streets. If one street passes through (6, 4) and (5, 2), what is the equation for a parallel street that passes through (−2, 6)?

A. y = 1 half x + 5
B. y - 1 half x + 1
C. y = 2x + 10
D. y = 2x − 14

Answer 1

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the first street.

`(\stackrel{x_1}{6}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{2}) ~\hfill~ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{2}-\stackrel{y1}{4}}}{\underset{\textit{\large run}} {\underset{x_2}{5}-\underset{x_1}{6}}} \implies \cfrac{ -2 }{ -1 } \implies 2`

so we are really looking for the equation of a line whose slope is 2 and it passes through (-2 , 6)

`(\stackrel{x_1}{-2}~,~\stackrel{y_1}{6})\hspace{10em} \stackrel{slope}{m} ~=~ 2 \\\\\\ \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-\stackrel{y_1}{6}=\stackrel{m}{2}(x-\stackrel{x_1}{(-2)}) \implies y -6 = 2 ( x +2) \\\\\\ y-6=2x+4\implies {\Large \begin{array}{llll} y=2x+10 \end{array}}`