Lines and slopes
Problem 13.309 · easy
Find the equation of the line through \( \displaystyle (-3, -3) \) that is parallel to \( \displaystyle y = - 4 x - 2 \), in slope-intercept form.
- Parallel lines share a slope. The slope is -4.Reviewed
- \[ - 4 \left(x + 3\right) - 3 = - 4 x - 15 \]Point-slope form, then simplify.✓ Proved
- \[ \left. - 4 x - 15 \right|_{\substack{ x=-3 }} = -3 \]The line passes through (-3, -3).✓ Proved
Answer \( y = - 4 x - 15 \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the slope from the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the point lies on the line.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the slope from the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the point lies on the line.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the slope from the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the point lies on the line.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-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 point lies on the line.gpt-oss:20b: fail (error) 2026-10-15 — The solution’s point‑slope step is incorrectly written and the final check is wrong: substituting x = –3 into y = –4x – 15 gives –27, not –3. The line does not pass through (–3,–3).
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.