Let's analyze the given problem step by step.
1. Understand the given line equation:
The equation of the lane passing through points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is given by:
[tex]\[ -7x + 3y = -21.5 \][/tex]
To find the slope of this line, we need to rearrange it into the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
[tex]\[ 3y = 7x - 21.5 \][/tex]
[tex]\[ y = \frac{7}{3}x - \frac{21.5}{3} \][/tex]
From the equation [tex]\( y = \frac{7}{3}x - \frac{21.5}{3} \)[/tex], we identify the slope [tex]\( m \)[/tex] as [tex]\( \frac{7}{3} \)[/tex].
2. Determine the slope of the central street [tex]\( PQ \)[/tex]:
- Parallel Lines:
If the central street [tex]\( PQ \)[/tex] is parallel to the given lane, it must have the same slope as the given line. So, the slope should be [tex]\( \frac{7}{3} \)[/tex] or equivalent.
- Perpendicular Lines:
If the central street [tex]\( PQ \)[/tex] is perpendicular to the given line, its slope must be the negative reciprocal of the slope of the given line. The negative reciprocal of [tex]\( \frac{7}{3} \)[/tex] is [tex]\( -\frac{3}{7} \)[/tex].
3. Match the given options with these slopes:
We have three options:
[tex]\[ \text{A. } -3x + 4y = 3 \][/tex]
[tex]\[ \text{B. } 3x + 7y = 63 \][/tex]
[tex]\[ \text{C. } 2x + y = 20 \][/tex]
We need to rearrange each option into the slope-intercept form and compare the slopes to the ones we calculated.
Option A:
[tex]\[ -3x + 4y = 3 \][/tex]
[tex]\[ 4y = 3x + 3 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{3}{4} \][/tex]
The slope here is [tex]\( \frac{3}{4} \)[/tex].
Option B:
[tex]\[ 3x + 7y = 63 \][/tex]
[tex]\[ 7y = -3x + 63 \][/tex]
[tex]\[ y = -\frac{3}{7}x + 9 \][/tex]
The slope here is [tex]\( -\frac{3}{7} \)[/tex].
Option C:
[tex]\[ 2x + y = 20 \][/tex]
[tex]\[ y = -2x + 20 \][/tex]
The slope here is [tex]\( -2 \)[/tex].
4. Comparison with target slopes:
- The slope for Option A is [tex]\( \frac{3}{4} \)[/tex], which does not match either [tex]\( \frac{7}{3} \)[/tex] or [tex]\( -\frac{3}{7} \)[/tex].
- The slope for Option B is [tex]\( -\frac{3}{7} \)[/tex], which matches the perpendicular slope.
- The slope for Option C is [tex]\( -2 \)[/tex], which does not match either [tex]\( \frac{7}{3} \)[/tex] or [tex]\( -\frac{3}{7} \)[/tex].
From our comparisons, the equation of the central street [tex]\( PQ \)[/tex] that fits the given conditions (being either parallel or perpendicular to the given lane) is:
[tex]\[ \boxed{\text{B. } 3x + 7y = 63} \][/tex]
