Which equation represents the new trail in the xy-plane?

Question

In a wildlife camera-trap study, a research team maps a reserve on a coordinate grid in which each unit represents 1 kilometer. An existing animal trail is represented by the line with equation 4x + 3y = 24. The team plans a new trail that passes through the camera station located at the point (6, 5) and is perpendicular to the existing trail.

Accepted Answer

A. y = (3/4)x + 1/2. Solving 4x + 3y = 24 for y gives y = -(4/3)x + 8, so the existing trail has slope -4/3. A perpendicular line has slope equal to the negative reciprocal, 3/4. Using (6, 5): 5 = (3/4)(6) + b, so b = 5 - 4.5 = 1/2, giving y = (3/4)x + 1/2. All four choices pass through (6, 5), so only the slope distinguishes them. B uses -3/4, the reciprocal without changing the sign. C uses 4/3, the sign change without taking the reciprocal. D uses -4/3, the slope of the original line (parallel instead of perpendicular).

Answer

B. y = -(3/4)x + 19/2 — You took the reciprocal of -4/3 but kept the negative sign; a perpendicular slope is the negative reciprocal, so it's +3/4.

Answer

C. y = (4/3)x - 3 — You flipped the sign of -4/3 but didn't take the reciprocal; the perpendicular slope is 3/4, not 4/3.

Answer

D. y = -(4/3)x + 13 — That's the slope of the existing trail itself, so this line is parallel to it, not perpendicular; you need the negative reciprocal, 3/4.

Algebra practice question

Answer: A. y = (3/4)x + 1/2

Why the other answers are wrong

More Algebra questions