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Lines and slopes

Problem 13.304 · easy

Find the equation of the line with slope \( \displaystyle - \frac{1}{3} \) through \( \displaystyle (1, -2) \), in slope-intercept form.

\[ \frac{1 - x}{3} - 2 = - \frac{x}{3} - \frac{5}{3} \]
Point-slope form, then simplify.✓ Proved
\[ \left. \frac{\left(-1\right) x}{3} - \frac{5}{3} \right|_{\substack{ x=1 }} = -2 \]
The line passes through (1, -2).✓ Proved

Answer \( y = - \frac{x}{3} - \frac{5}{3} \)

Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the line meets the point with the required slope

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly derives the equation and verifies it passes through the given point with the correct slope.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly derives the equation and verifies it passes through the given point with the correct slope.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the slope-intercept form and verifies the point. The steps are logically sound and algebraically correct.
gpt-oss:20b: fail (error) 2026-10-16 — The first sentence claims to use point‑slope form but omits the variable y; it should read "y + 2 = -(x-1)/3" before simplifying to the given equation. This omission makes the setup unclear and could mislead a student.
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are sound.
gpt-oss:20b: pass 2026-10-15

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/lines_slopes, checked 2026-10-18 with SymPy 1.14.0.