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Lines and slopes

Problem 13.138 · easy

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

\[ 3 - \frac{2 \left(x + 5\right)}{3} = - \frac{2 x}{3} - \frac{1}{3} \]
Point-slope form, then simplify.✓ Proved
\[ \left. - \frac{2 x}{3} - 1 \cdot \frac{1}{3} \right|_{\substack{ x=-5 }} = 3 \]
The line passes through (-5, 3).✓ Proved

Answer \( y = - \frac{2 x}{3} - \frac{1}{3} \)

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 steps are logically sound and algebraically correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly derives the slope-intercept form and verifies the point. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly derives the slope-intercept form and verifies the point. The steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the slope-intercept form and verifies the point. Although the intermediate steps are condensed into a single line, the algebraic result is correct and the verification holds.
gpt-oss:20b: fail (misleading) 2026-10-14 — The first sentence claims to present the point‑slope form, but it actually equates two algebraic expressions rather than writing the line as y = … . It misleads the reader into thinking the equation of the line has been derived, when in fact the correct form y = -2x/3 - 1/3 is never explicitly stated. The second sentence merely checks a point, which is fine, but the overall solution is incomplete and confusing.

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.