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

Can we have a picture of the Mapping? It would help tremendously.

Answer 2

okay.....PQ is perpendicular to AB

Now we will look at the line given..-7x + 3y = -21.5
we need to find the slope, so we will put the equation in y = mx + b form, where the number in the m position is the slope.
-7x + 3y = - 21.5 --- add 7x to both sides
3y = 7x - 21.5 -- now divide both sides by 3
y = 7/3x - 21.5/3
the slope (m) is 7/3
However, we are looking for a perpendicular line, so we need the negative reciprocal slope. All that means is " flip " the slope and change the sign.
So the slope we need is -3/7. (see how I flipped the slope and changed the sign.

Notice that PQ is going through points (7,6)....we will use this.
Now we use y = mx + b
slope(m) = -3/7
(7,6)....x = 7 and y = 6
now we sub
y = mx + b
6 = -3/7(7) + b
6 = -3 + b
6 + 3 = b
9 = b

so our equation of the central street PQ is : y = -3/7x + 9

and if you need it in standard form :
Ax + By = C

y = -3/7x + 9 --- multiply everything by 7
7y = -3x + 9 --- add 3 to both sides
3x + 7y = 9 <== in standard form