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Lines and slopes

Problem 13.310 · easy

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

\[ 2 - \frac{3 \left(x - 5\right)}{2} = \frac{19}{2} - \frac{3 x}{2} \]
Point-slope form, then simplify.✓ Proved
\[ \left. - \frac{3 x}{2} + \frac{19}{2} \right|_{\substack{ x=5 }} = 2 \]
The line passes through (5, 2).✓ Proved

Answer \( y = \frac{19}{2} - \frac{3 x}{2} \)

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 (error) — The solution fails to explicitly state the point-slope formula or the substitution step used to derive the equation. It presents an algebraic identity and a verification check without showing the logical derivation from the given slope and point to the final slope-intercept form.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly state the point-slope formula or the substitution step used to derive the equation. It presents an algebraic identity and a verification check without showing the logical derivation from the given slope and point to the final slope-intercept form.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the slope-intercept form and verifies the point. The steps are algebraically sound and logically complete.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution provides no derivation or explanation of how the slope-intercept form was obtained from the given point and slope. It merely verifies the final answer algebraically, failing to demonstrate the required method (using y - y1 = m(x - x1)).
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.