Lines and slopes
Problem 13.128 · easy
Find the equation of the line through \( \displaystyle (2, 1) \) that is perpendicular to \( \displaystyle y = 5 x \), in slope-intercept form.
- Perpendicular slopes are negative reciprocals: -1/5.
- \[ 1 - \frac{x - 2}{5} = \frac{7}{5} - \frac{x}{5} \]Point-slope form, then simplify.✓ Proved
- \[ \left. - \frac{x}{5} + \frac{7}{5} \right|_{\substack{ x=2 }} = 1 \]The line passes through (2, 1).✓ Proved
Answer \( y = \frac{7}{5} - \frac{x}{5} \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 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: fail (error) — Sentence 2 incorrectly presents the line equation as an identity 1-(x-2)/5 = 7/5-x/5 instead of y = 1-(x-2)/5 (or y = 7/5 - x/5). The step omits the variable y and misstates the point‑slope form.qwen3.6:27b-mlx: pass — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The steps are logically sound and the final answer matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The steps are logically sound and the final answer matches the stated answer.gpt-oss:20b: fail (error) 2026-10-17 — Sentence 2 incorrectly presents the line equation as an identity 1-(x-2)/5 = 7/5-x/5 instead of y = 1-(x-2)/5 (or y = 7/5 - x/5). The step omits the variable y and misstates the point‑slope form.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The steps are logically sound and the final answer matches the stated answer.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The algebraic steps are valid.gpt-oss:20b: fail (error) 2026-10-14 — The solution incorrectly writes the line equation as an equality of two expressions instead of expressing y in terms of x. The point‑slope form should be y‑1 = (-1/5)(x‑2), leading to y = 7/5 – x/5. The current presentation could mislead a student about how to write the equation of a line.
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.