Let's solve the given problem step by step.
Given the equation of the lane passing through points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is:
[tex]\[
-7x + 3y = -21.5
\][/tex]
We need to find the equation of the central street PQ from the given choices, knowing that it should be perpendicular to the lane passing through [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
1. Convert the given equation to slope-intercept form [tex]\(y = mx + b\)[/tex]:
The given equation is:
[tex]\[
-7x + 3y = -21.5
\][/tex]
To convert this to the slope-intercept form [tex]\(y = mx + b\)[/tex], solve for [tex]\(y\)[/tex]:
[tex]\[
3y = 7x - 21.5
\][/tex]
Divide every term by 3:
[tex]\[
y = \frac{7}{3}x - \frac{21.5}{3}
\][/tex]
From this equation, the slope [tex]\(m\)[/tex] of the lane passing through [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is:
[tex]\[
m = \frac{7}{3}
\][/tex]
2. Find the slope of the line perpendicular to the given lane:
For two lines to be perpendicular, the product of their slopes must be -1. Therefore, the slope of the perpendicular line to the given lane is the negative reciprocal of [tex]\(\frac{7}{3}\)[/tex]:
[tex]\[
\text{Perpendicular slope} = -\frac{1}{\frac{7}{3}} = -\frac{3}{7}
\][/tex]
3. Compare the slopes with the given equations to find the matching one:
Check each given option to see which one has a slope of [tex]\(-\frac{3}{7}\)[/tex]:
- Option A:
[tex]\[
-3x + 4y = 3
\][/tex]
Rewrite in slope-intercept form:
[tex]\[
4y = 3x + 3
\][/tex]
[tex]\[
y = \frac{3}{4}x + \frac{3}{4}
\][/tex]
The slope here is [tex]\(\frac{3}{4}\)[/tex], which is not what we need.
- Option B:
[tex]\[
3x + 7y = 63
\][/tex]
Rewrite in slope-intercept form:
[tex]\[
7y = -3x + 63
\][/tex]
[tex]\[
y = -\frac{3}{7}x + 9
\][/tex]
The slope here is [tex]\(-\frac{3}{7}\)[/tex], which matches the perpendicular slope we are looking for.
- Option C:
[tex]\[
2x + y = 20
\][/tex]
Rewrite in slope-intercept form:
[tex]\[
y = -2x + 20
\][/tex]
The slope here is [tex]\(-2\)[/tex], which is not what we need.
- Option D:
[tex]\[
7x + 3y = 70
\][/tex]
Rewrite in slope-intercept form:
[tex]\[
3y = -7x + 70
\][/tex]
[tex]\[
y = -\frac{7}{3}x + \frac{70}{3}
\][/tex]
The slope here is [tex]\(-\frac{7}{3}\)[/tex], which is not what we need.
Hence, the correct option is:
[tex]\[
\boxed{B}
\][/tex]
