Lines and slopes
Problem 13.129 · easy
Find the equation of the line through \( \displaystyle (1, -2) \) that is parallel to \( \displaystyle y = x \), in slope-intercept form.
- Parallel lines share a slope. The slope is 1.
- \[ 1 \left(x - 1\right) - 2 = x - 3 \]Point-slope form, then simplify.✓ Proved
- \[ \left. x - 3 \right|_{\substack{ x=1 }} = -2 \]The line passes through (1, -2).✓ Proved
Answer \( y = x - 3 \)
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 applies point‑slope form; it omits the variable y and equates the left side to the right side directly. The correct derivation is y+2 = 1·(x−1) ⇒ y = x−3.qwen3.6:27b-mlx: pass — The solution correctly identifies the slope and derives the correct equation. The intermediate step is algebraically valid, though the notation in line 2 is slightly unconventional.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the slope and derives the correct equation. The intermediate step is algebraically valid, though the notation in line 2 is slightly unconventional.gpt-oss:20b: fail (error) 2026-10-17 — Sentence 2 incorrectly applies point‑slope form; it omits the variable y and equates the left side to the right side directly. The correct derivation is y+2 = 1·(x−1) ⇒ y = x−3.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the slope and uses the point-slope form to derive the correct equation. The steps are logically sound and the final answer is correct.gpt-oss:20b: fail (error) 2026-10-14 — Sentence 2 incorrectly applies the point‑slope formula: it omits the variable y and mixes terms. The correct derivation is y+2=1·(x−1) ⇒ y=x−3. The subsequent check is fine, but the earlier step is algebraically incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly applies the point-slope formula. It writes `1*(x - 1) - 2 = x - 3`, which implies `y - 2` instead of `y - (-2)`. The correct derivation is `y - (-2) = 1(x - 1)`, leading to `y = x - 3`. While the final answer is correct, the intermediate step contains a sign error that would teach a student incorrect algebra.gpt-oss:20b: fail (error) 2026-10-14 — Sentence 2 incorrectly applies point‑slope form; it does not produce the equation y=x−3 and misrepresents the relationship between y and x.
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.