Question

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 A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?

A.
-3x + 4y = 3

B.
3x + 7y = 63

C.
2x + y = 20

D.
7x + 3y = 70

Answer 1

B. 3x + 7y = 63

Explanation:

to get the slope-intercept form of our AB line (to see slope and y-intercept directly), we need to get it into a

y = ax + b

form. "a" is the slope, "b" the y-intercept.

-7x + 3y = -21.5

3y = 7x - 21.5

y = (7/3)x - 21.5/3

the slope of AB is 7/3.

the slope of a perpendicular line turns this ratio upside down and flips the sign.

so, the slope of PQ is -3/7.

and its slope-intercept form is

y = (-3/7)x + b

and therefore the equation form used in the answer options

7y = -3x + b

3x + 7y = b

the only answer option that fits is

B. 3x + 7y = 63

Answer 2

To find the equation of the central street PQ, we need to determine the slope of the given lane passing through points A and B.

The equation -7x + 3y = -21.5 can be rewritten in slope-intercept form (y = mx + b) by solving for y:

3y = 7x - 21.5

y = (7/3)x - 7.17

The slope of the given lane is 7/3.

For the central street PQ to be perpendicular to the given lane, its slope must be the negative reciprocal of 7/3, which is -3/7.

Therefore, the equation of the central street PQ can be written in the point-slope form using point P (let's assume it has coordinates (x1, y1)):

y - y1 = -3/7(x - x1)

Since the equation does not provide the coordinates of point P, we cannot determine the exact equation of the central street PQ.

However, we can eliminate the answer choices that do not have a slope of -3/7. Checking the slopes of the given answer choices:

A. -3x + 4y = 3 -> slope = 3/4 (not -3/7)

B. 3x + 7y = 63 -> slope = -3/7 (matches the slope we need)

C. 2x + y = 20 -> slope = -2 (not -3/7)

D. 7x + 3y = 70 -> slope = -7/3 (not -3/7)

Based on this analysis, the equation that matches the slope we need (-3/7) is answer choice B: 3x + 7y = 63.