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? A. -3x + 4y = 3 B. -1.5x − 3.5y = -31.5 C. 2x + y = 20 D. -2.25x + y = -9.75

  B.  -1.5x − 3.5y = -31.5

Step-by-step explanation:

You want an equation of the line perpendicular to -7x +3y = -21.5 that goes through the point (7, 6).

When the equation of the reference line is given in this form, the equation of a perpendicular line can be found by swapping the x- and y-coefficients, and negating one of them. Swapping coefficients, we have ...

  3x -7y = constant

Negating the x-coefficient gives ...

  -3x -7y = constant

Filling in the given point values, we can find the constant:

  -3(7) -7(6) = constant = -21 -42 = -63

None of the answer choices matches this, but one does match when we divide the numbers by 2:

  -3x -7y = -63

  -1.5x -3.5y = -31.5 . . . . . . matches choice B

y = (-3/7)x + 9

Step-by-step explanation:

IF AB has the equation -7x + 3y = -21.5, then 3y = 7x - 21.5 is equivalent.  Dividing both sides by 3 yields y = (7/3)x - 21.5/3.  This reveals that the slope of AB is 7/3 and that the slope of any line perpendicular to AB, as central street PQ is, is the negative reciprocal of the slope of AB, or -3/7.

Thus PQ, being perpendicular to AB, has the slope -3/7.  Starting with y = mx + b, and using the coordinates of P(7, 6), we get:

6 = (-3/7)(7) + b, or 9 = b, so that y = (-3/7)x + 9.