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. 3x + 7y = 63
C. 2x + y = 20
D. 7x + 3y = 70
To find the equation of the central street PQ, we need to determine its relationship to the given street AB, which has the equation -7x + 3y = -21.5.
First, we can rearrange the equation of street AB into slope-intercept form (y = mx + b) to find its slope:
1. Rewrite the equation:
( 3y = 7x – 21.5 )
2. Divide by 3:
( y = frac{7}{3}x – frac{21.5}{3} )
The slope ( m ) of street AB is ( frac{7}{3} ). The central street PQ must be perpendicular to AB; thus, its slope is the negative reciprocal of ( frac{7}{3} ), which is ( -frac{3}{7} ).
Now, we can check each option for an equation that fits this slope:
A. -3x + 4y = 3
Rearranging gives:
( 4y = 3x + 3 )
( y = frac{3}{4}x + frac{3}{4} ) (slope is ( frac{3}{4} ))
B. 3x + 7y = 63
Rearranging gives:
( 7y = -3x +