Question

A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?

Answer 1

The equation of the line passing through A and B is

$-7x+3y=-21.5$

This could be written as

$y=\frac{7}{3}x-\frac{21.5}{3}$

By comparing with the standard form $y=mx+c$

The slope $m=\frac{7}{3}$

Central street PQ will be perpendicular to the lane passing through A and B

Product of slope of perpendicular lines will be equal to -1

$m_1.m_2=-1$

slope of perpendicular line

$=\frac{-1}{slope\space parallel\space line}=\frac{-1}{\frac{7}{3}}=\frac{-3}{7}$

This is the slope of the line.

Thus the equation of line will be $y=mx+c$

$y=\frac{-3}{7}x+c$

$7y=-3x+7c\\7y+3x=7c$

dividing by 2

$3.5y+1.5x=3.5c$

To find c with this information is impossible

To check for the line with slope as $\frac{-3}{7}$ OP in a similar figure is found

The line PQ passes through the point(7,6) in the figure above hence the line could be found out using point slope form of line.

$y-y_1=m(x-x_1)$

$slope=\frac{-3}{7} (7,6)$

$y-6=\frac{-3}{7}(x-7)\implies7(y-6)=-3(x-7)\\7y-42=-3x+21\implies7y+3x=21+42$

$7y+3x=63$ is the equation of central lane

Dividing by 2

$\implies3.5y-1.5x=-31.5$ is also the equation.