To determine which of the given options represents the equation of the central street [tex]\(PQ\)[/tex], let's start by analyzing the equation of the lane passing through points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
Given equation: [tex]\(-7x + 3y = -21.5\)[/tex]
To understand this equation better, let's find its slope. We can rearrange it into the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope.
[tex]\[
-7x + 3y = -21.5
\][/tex]
- Isolate [tex]\(y\)[/tex]:
[tex]\[
3y = 7x - 21.5
\][/tex]
[tex]\[
y = \frac{7}{3}x - \frac{21.5}{3}
\][/tex]
The slope [tex]\(m\)[/tex] of the line is [tex]\(\frac{7}{3}\)[/tex].
Now, we need to consider that the streets are either going to be parallel or perpendicular to this given lane.
1. If a line is parallel to the given lane, it will have the same slope [tex]\(\frac{7}{3}\)[/tex].
2. If a line is perpendicular to the given lane, its slope will be the negative reciprocal of [tex]\(\frac{7}{3}\)[/tex], which is [tex]\(-\frac{3}{7}\)[/tex].
Let's evaluate the slopes of the lines provided in the options:
### Option A: [tex]\(-3x + 4y = 3\)[/tex]
- Isolate [tex]\(y\)[/tex]:
[tex]\[
4y = 3x + 3
\][/tex]
[tex]\[
y = \frac{3}{4}x + \frac{3}{4}
\][/tex]
The slope [tex]\(m\)[/tex] of this line is [tex]\(\frac{3}{4}\)[/tex], which is neither [tex]\(\frac{7}{3}\)[/tex] nor [tex]\(-\frac{3}{7}\)[/tex]. Therefore, this option is not correct.
### Option B: [tex]\(3x + 7y = 63\)[/tex]
- Isolate [tex]\(y\)[/tex]:
[tex]\[
7y = -3x + 63
\][/tex]
[tex]\[
y = -\frac{3}{7}x + 9
\][/tex]
The slope [tex]\(m\)[/tex] of this line is [tex]\(-\frac{3}{7}\)[/tex], which is the negative reciprocal of [tex]\(\frac{7}{3}\)[/tex]. This indicates that this line is perpendicular to the given lane. This option is correct.
### Option C: [tex]\(2x + y = 20\)[/tex]
- Isolate [tex]\(y\)[/tex]:
[tex]\[
y = -2x + 20
\][/tex]
The slope [tex]\(m\)[/tex] of this line is [tex]\(-2\)[/tex], which is neither [tex]\(\frac{7}{3}\)[/tex] nor [tex]\(-\frac{3}{7}\)[/tex]. Therefore, this option is not correct.
### Option D: [tex]\(7x + 3y = 70\)[/tex]
- Isolate [tex]\(y\)[/tex]:
[tex]\[
3y = -7x + 70
\][/tex]
[tex]\[
y = -\frac{7}{3}x + \frac{70}{3}
\][/tex]
The slope [tex]\(m\)[/tex] of this line is [tex]\(-\frac{7}{3}\)[/tex], which is not relevant for our parallel or perpendicular condition. Therefore, this option is not correct.
Therefore, the correct equation for the central street [tex]\(PQ\)[/tex] is:
[tex]\[
\boxed{3x + 7y = 63}
\][/tex]
