Let's solve the problem step-by-step.
Given the equation of the lane passing through points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ -7x + 3y = -21.5 \][/tex]
We need to compare the slopes of each of the given options to find out if they are perpendicular or parallel to this street.
First, we convert the given equation of the lane into the slope-intercept form, [tex]\(y = mx + b\)[/tex], to find its slope [tex]\(m\)[/tex].
1. Convert the given equation to slope-intercept form:
[tex]\[ -7x + 3y = -21.5 \][/tex]
[tex]\[ 3y = 7x - 21.5 \][/tex]
[tex]\[ y = \frac{7}{3}x - \frac{21.5}{3} \][/tex]
Therefore, the slope [tex]\(m\)[/tex] of the given line is:
[tex]\[ m = \frac{7}{3} \][/tex]
2. Examine each of the given options to determine their slopes:
Option A: [tex]\( -3x + 4y = 3 \)[/tex]
[tex]\[ 4y = 3x + 3 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{3}{4} \][/tex]
Therefore, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = \frac{3}{4} \][/tex]
Option B: [tex]\( 3x + 7y = 63 \)[/tex]
[tex]\[ 7y = -3x + 63 \][/tex]
[tex]\[ y = -\frac{3}{7}x + \frac{63}{7} \][/tex]
Therefore, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = -\frac{3}{7} \][/tex]
Option C: [tex]\( 2x + y = 20 \)[/tex]
[tex]\[ y = -2x + 20 \][/tex]
Therefore, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = -2 \][/tex]
Option D: [tex]\( 7x + 3y = 70 \)[/tex]
[tex]\[ 3y = -7x + 70 \][/tex]
[tex]\[ y = -\frac{7}{3}x + \frac{70}{3} \][/tex]
Therefore, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = -\frac{7}{3} \][/tex]
3. Determine which of the options results in either parallel or perpendicular lines:
- Parallel lines have the same slope.
- Perpendicular lines have slopes that are negative reciprocals of each other; that is, the product of their slopes is [tex]\(-1\)[/tex].
Given slope: [tex]\(\frac{7}{3}\)[/tex]
Checking for perpendicular:
- The negative reciprocal of [tex]\(\frac{7}{3}\)[/tex] is [tex]\(-\frac{3}{7}\)[/tex].
From the options, only option B has the slope [tex]\(-\frac{3}{7}\)[/tex], which is the negative reciprocal of [tex]\(\frac{7}{3}\)[/tex].
Hence, the equation of the central street [tex]\(PQ\)[/tex] that is perpendicular to the given lane is:
[tex]\[ \boxed{3x + 7y = 63} \][/tex]
