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Lines and slopes

Problem 13.499 · easy

Find the equation of the line through \( \displaystyle (-2, 2) \) that is perpendicular to \( \displaystyle y = - 4 x - 4 \), in slope-intercept form.

\[ - \frac{1}{-4} = \frac{1}{4} \]
Perpendicular slopes are negative reciprocals.✓ Proved
\[ \frac{x + 2}{4} + 2 = \frac{x}{4} + \frac{5}{2} \]
Point-slope form, then simplify.✓ Proved
\[ \left. \frac{x}{4} + \frac{5}{2} \right|_{\substack{ x=-2 }} = 2 \]
The line passes through (-2, 2).✓ Proved

Answer \( y = \frac{x}{4} + \frac{5}{2} \)

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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
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 verifies the point satisfies the resulting equation. The steps are logically sound and algebraically correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly identifies the negative reciprocal slope and verifies the point satisfies the resulting equation. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-11-13
qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly identifies the negative reciprocal slope and verifies the point-slope form. The steps are logically sound and the final answer is correct.
gpt-oss:20b: pass 2026-11-13

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 2026-11-13 with SymPy 1.14.0.