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Lines and slopes

Problem 13.133 · easy

Find the equation of the line with slope \( \displaystyle 2 \) through \( \displaystyle (-2, -2) \), in slope-intercept form.

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

Answer \( y = 2 x + 2 \)

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: fail (misleading) — The solution lacks the crucial step of explicitly stating the point-slope formula and substituting the given values to derive the equation. It jumps directly to an algebraic identity without showing the modeling step, which is the core of the problem.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution lacks the crucial step of explicitly stating the point-slope formula and substituting the given values to derive the equation. It jumps directly to an algebraic identity without showing the modeling step, which is the core of the problem.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are valid and the final answer is correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution lacks the crucial step of stating the point-slope formula and substituting the given values to derive the equation. It jumps directly to an algebraic identity and a verification step, failing to demonstrate the derivation process required by the problem.
gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope equation is written incorrectly: it should be \(y+2=2(x+2)\), not \(2(x+2)-2=2x+2\)."}

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 2026-10-17 with SymPy 1.14.0.