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Lines and slopes

Problem 13.347 · easy

Find the equation of the line with slope \( \displaystyle - \frac{5}{3} \) through \( \displaystyle (-2, 2) \), in slope-intercept form.

\[ 2 - \frac{5 \left(x + 2\right)}{3} = - \frac{5 x}{3} - \frac{4}{3} \]
Point-slope form, then simplify.✓ Proved
\[ \left. - \frac{5 x}{3} - \frac{4}{3} \right|_{\substack{ x=-2 }} = 2 \]
The line passes through (-2, 2).✓ Proved

Answer \( y = - \frac{5 x}{3} - \frac{4}{3} \)

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 slope-intercept form and verifies the point. The algebraic steps are sound and the final answer matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are sound and the final answer matches the stated answer.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the slope-intercept form from the point-slope form and verifies the result. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution is labeled as 'unchecked', but the prompt instructions state that equations are checked by a CAS. More importantly, the solution lacks the necessary logical steps to derive the equation from the problem statement; it merely verifies the final answer rather than showing the derivation (e.g., using y - y1 = m(x - x1)). A student would not learn how to find the equation, only how to check it.
gpt-oss:20b: fail (error) 2026-10-16 — The first line omits the dependent variable y in the point‑slope form, writing "2 - 5*(x + 2)/3 = …" instead of "y - 2 = -5/3 (x + 2)", so the setup is not correctly expressed.

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.