To find the equation of the central street [tex]\(PQ\)[/tex], we need to consider the relationship between the given line [tex]\(AB\)[/tex] and the line [tex]\(PQ\)[/tex]. Here is a step-by-step solution:
1. Find the slope of the line passing through [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
The equation of the line is given as:
[tex]\[
-7x + 3y = -21.5
\][/tex]
To find the slope, we convert this equation to the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Starting from the given equation:
[tex]\[
-7x + 3y = -21.5
\][/tex]
Add [tex]\(7x\)[/tex] to both sides:
[tex]\[
3y = 7x - 21.5
\][/tex]
Divide every term by 3:
[tex]\[
y = \frac{7}{3}x - \frac{21.5}{3}
\][/tex]
The slope of the line [tex]\(AB\)[/tex] (denoted as [tex]\(m_{AB}\)[/tex]) is:
[tex]\[
m_{AB} = \frac{7}{3}
\][/tex]
2. Determine the slope of the perpendicular line [tex]\(PQ\)[/tex]:
The central street [tex]\(PQ\)[/tex] is perpendicular to the street [tex]\(AB\)[/tex]. The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, the slope of [tex]\(PQ\)[/tex] (denoted as [tex]\(m_{PQ}\)[/tex]) is:
[tex]\[
m_{PQ} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{7}{3}} = -\frac{3}{7}
\][/tex]
3. Identify the equation of the central street from the provided options:
We need to find which of the given options has a slope of [tex]\(-\frac{3}{7}\)[/tex]. Let's convert each option to slope-intercept form [tex]\(y = mx + b\)[/tex] and determine its slope:
- Option A: [tex]\(-3x + 4y = 3\)[/tex]
[tex]\[
4y = 3x + 3 \implies y = \frac{3}{4}x + \frac{3}{4}
\][/tex]
The slope is [tex]\( \frac{3}{4} \)[/tex].
- Option B: [tex]\(3x + 7y = 63\)[/tex]
[tex]\[
7y = -3x + 63 \implies y = -\frac{3}{7}x + 9
\][/tex]
The slope is [tex]\( -\frac{3}{7} \)[/tex].
- Option C: [tex]\(2x + y = 20\)[/tex]
[tex]\[
y = -2x + 20
\][/tex]
The slope is [tex]\( -2 \)[/tex].
- Option D: [tex]\(7x + 3y = 70\)[/tex]
[tex]\[
3y = -7x + 70 \implies y = -\frac{7}{3}x + \frac{70}{3}
\][/tex]
The slope is [tex]\( -\frac{7}{3} \)[/tex].
4. Compare the slopes:
From our calculations, we see that the slope of Option B matches the required slope of [tex]\(-\frac{3}{7}\)[/tex].
Therefore, the equation of the central street [tex]\(PQ\)[/tex] is given by:
[tex]\[
\boxed{3x + 7y = 63}
\][/tex]
