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 simplest mathematical model for relating two variables is the linear equation in two variables. We can write this equation in The Slope-Intercept Form as follows:

$y=mx+b$

The equation is called linear because its graph is a line. (In mathematics, the term line means straight line.

So a software designer is mapping the streets for a new racing game. We know that the equation of the line passing through A and B is:

$-7x+3y=-21.5$

Next, let's write it in The Slope-Intercept Form:

$y=\frac{7}{3}x-\frac{43}{6}$

We know that the line passing through AB is perpendicular to the line passing through PQ. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is:

$m_{1}=-\frac{1}{m_{2}}$

Therefore, the slope for the line passing through PQ is:

$m_{PQ}=\frac{-1}{\frac{7}{3}}=-\frac{3}{7}$

and the point $P$ is:

$P(7,6)$

Finally, using the Point-Slope Form we can get the equation of the central street PQ:

$y-y_{1}=m_{PQ}(x-x1) \\ \\ y-6=-\frac{3}{7}(x-7) \\ \\ 7y-42=-3x+21 \\ \\ \boxed{7y+3x=63}$