Lines and slopes
Problem 13.135 · easy
Find the equation of the line through \( \displaystyle (4, -2) \) that is parallel to \( \displaystyle y = 4 x - 1 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 4.
- \[ 4 \left(x - 4\right) - 2 = 4 x - 18 \]Point-slope form, then simplify.✓ Proved
- \[ \left. 4 x - 18 \right|_{\substack{ x=4 }} = -2 \]The line passes through (4, -2).✓ Proved
Answer \( y = 4 x - 18 \)
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 (misleading) — The second sentence incorrectly applies point‑slope form: it writes 4*(x-4)-2 = 4*x-18, which is not the correct derivation of the line’s equation. The correct step is y+2 = 4(x-4) leading to y = 4x-18.qwen3.6:27b-mlx: pass — The solution correctly identifies the slope and verifies the point. Although line 2 is labeled as an equation check rather than a derivation step, the logic is sound and the final answer is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the slope and verifies the point. Although line 2 is labeled as an equation check rather than a derivation step, the logic is sound and the final answer is correct.gpt-oss:20b: fail (misleading) 2026-10-17 — The second sentence incorrectly applies point‑slope form: it writes 4*(x-4)-2 = 4*x-18, which is not the correct derivation of the line’s equation. The correct step is y+2 = 4(x-4) leading to y = 4x-18.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the slope of the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the result.gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope step is incorrect: it should be \(y+2=4(x-4)\), not \(4(x-4)-2=4x-18\)."}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 verification step confirms the point lies on the line.gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope form is misapplied: it should be \(y+2=4(x-4)\), not \(4(x-4)-2=4x-18\)."}
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.