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Lines and slopes

Problem 13.306 · easy

Find the equation of the line through \( \displaystyle (-3, 3) \) that is perpendicular to \( \displaystyle y = 3 x - 3 \), in slope-intercept form.

Perpendicular slopes are negative reciprocals: -1/3.
\[ 3 - \frac{x + 3}{3} = 2 - \frac{x}{3} \]
Point-slope form, then simplify.✓ Proved
\[ \left. 2 - \frac{x}{3} \right|_{\substack{ x=-3 }} = 3 \]
The line passes through (-3, 3).✓ Proved

Answer \( y = 2 - \frac{x}{3} \)

Lines: 2 proved, 1 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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: fail (error) — The solution fails to explicitly state the point-slope equation used to derive the final answer. It jumps from identifying the slope to a simplified algebraic identity, omitting the crucial step of writing y - 3 = -1/3(x + 3) or y = -1/3(x + 3) + 3, which is necessary to demonstrate how the line equation was constructed.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly state the point-slope equation used to derive the final answer. It jumps from identifying the slope to a simplified algebraic identity, omitting the crucial step of writing y - 3 = -1/3(x + 3) or y = -1/3(x + 3) + 3, which is necessary to demonstrate how the line equation was constructed.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the negative reciprocal slope and verifies the point satisfies the derived equation. The logic is sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution claims to use point-slope form in step 2 but presents an algebraic identity (3 - (x + 3)/3 = 2 - x/3) rather than the equation of the line. It fails to explicitly state the equation y - 3 = -1/3(x + 3) or its simplified form, making the derivation opaque and the 'simplify' step unjustified.
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.