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A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is [tex]\(-7x + 3y = -21.5\)[/tex]. What is the equation of the central street [tex]\(PQ\)[/tex]?
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].
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