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Lines and slopes

Problem 13.348 · easy

Find the equation of the line through \( \displaystyle (-3, -3) \) that is perpendicular to \( \displaystyle y = 4 x + 1 \), in slope-intercept form.

Perpendicular slopes are negative reciprocals: -1/4.
Reviewed
\[ \frac{- x - 3}{4} - 3 = - \frac{x}{4} - \frac{15}{4} \]
Point-slope form, then simplify.✓ Proved
\[ \left. \frac{\left(-1\right) x}{4} - \frac{15}{4} \right|_{\substack{ x=-3 }} = -3 \]
The line passes through (-3, -3).✓ Proved

Answer \( y = - \frac{x}{4} - \frac{15}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The verification step confirms the point lies on the line.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The verification step confirms the point lies on the line.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The verification step confirms the point lies on the line.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The verification step confirms the point lies on the line.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.