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 line passing through a and b is -7x + 3y = -21.5. what is the equation of the central street pq? a. -3x + 4y = 3 b. 3x + 7y = 63 c. 2x + y = 20 d. 7x + 3y = 70

Explanation:

Step1: Find slope of AB

First, rewrite $-7x + 3y = -21.5$ to slope-intercept form $y=mx+b$:
$$3y=7x-21.5 \implies y=\frac{7}{3}x-\frac{21.5}{3}$$
Slope of AB: $m_{AB}=\frac{7}{3}$

Step2: Find slope of PQ

PQ is perpendicular to AB, so slope $m_{PQ}=-\frac{3}{7}$ (negative reciprocal)

Step3: Use point P(7,6) to find b

Substitute $x=7$, $y=6$, $m=-\frac{3}{7}$ into $y=mx+b$:
$$6=-\frac{3}{7}(7)+b \implies 6=-3+b \implies b=9$$

Step4: Convert to standard form

$y=-\frac{3}{7}x+9$, multiply by 7:
$$7y=-3x+63 \implies 3x+7y=63$$

Answer:

B. $3x + 7y = 63$