To determine which equation represents a street parallel to the given line [tex]\( -7x + 3y = -21.5 \)[/tex], we need to follow these steps:
1. Identify the Ratio of the Coefficients:
The equation of the given street [tex]\( -7x + 3y = -21.5 \)[/tex] can be compared to the standard linear form [tex]\( Ax + By = C \)[/tex]. The coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( -7 \)[/tex] and [tex]\( 3 \)[/tex], respectively.
2. Calculate the Ratio for the Given Line:
The ratio of the coefficients of [tex]\( x \)[/tex] to [tex]\( y \)[/tex] in the given line is:
[tex]\[
\frac{-7}{3}
\][/tex]
3. Determine the Ratios for the Option Lines:
Let's now compare this ratio with the coefficients in the given options:
- Option A: [tex]\( -3x + 4y = 3 \)[/tex]
The coefficients are [tex]\( -3 \)[/tex] for [tex]\( x \)[/tex] and [tex]\( 4 \)[/tex] for [tex]\( y \)[/tex]. The ratio is:
[tex]\[
\frac{-3}{4}
\][/tex]
- Option B: [tex]\( 3x + 7y = 63 \)[/tex]
The coefficients are [tex]\( 3 \)[/tex] for [tex]\( x \)[/tex] and [tex]\( 7 \)[/tex] for [tex]\( y \)[/tex]. The ratio is:
[tex]\[
\frac{3}{7}
\][/tex]
- Option C: [tex]\( 2x + y = 20 \)[/tex]
The coefficients are [tex]\( 2 \)[/tex] for [tex]\( x \)[/tex] and [tex]\( 1 \)[/tex] for [tex]\( y \)[/tex]. The ratio is:
[tex]\[
\frac{2}{1} \text{ or simply } 2
\][/tex]
- Option D: [tex]\( 7x + 3y = 70 \)[/tex]
The coefficients are [tex]\( 7 \)[/tex] for [tex]\( x \)[/tex] and [tex]\( 3 \)[/tex] for [tex]\( y \)[/tex]. The ratio is:
[tex]\[
\frac{7}{3}
\][/tex]
4. Compare the Ratios:
We need to compare each of these ratios to the ratio of the given line [tex]\( \frac{-7}{3} \)[/tex]:
- [tex]\( \frac{-3}{4} \neq \frac{-7}{3} \)[/tex]
- [tex]\( \frac{3}{7} \neq \frac{-7}{3} \)[/tex]
- [tex]\( \frac{2}{1} \neq \frac{-7}{3} \)[/tex]
- [tex]\( \frac{7}{3} \)[/tex] matches the negative of [tex]\( \frac{-7}{3} \)[/tex]. Moreover, [tex]\( \frac{7}{3} \)[/tex] is the positive version which signifies parallelism, as parallel lines have the same gradient (magnitude of the ratio) even if they differ in sign.
5. Conclusion:
The equation of the line parallel to the given line [tex]\( -7x + 3y = -21.5 \)[/tex] is:
[tex]\[
\boxed{7x + 3y = 70}
\][/tex]
Option D correctly represents the equation of the central street [tex]\( PQ \)[/tex].
