Lines and slopes
Problem 13.423 · easy
Find the equation of the line through \( \displaystyle (2, -1) \) that is parallel to \( \displaystyle y = x - 1 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 1.Reviewed
- \[ 1 \left(x - 2\right) - 1 = x - 3 \]Point-slope form, then simplify.✓ Proved
- \[ \left. x - 3 \right|_{\substack{ x=2 }} = -1 \]The line passes through (2, -1).✓ Proved
Answer \( y = x - 3 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 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.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-22 — 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: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-21 — 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 2026-10-21qwen3.6:27b-mlx: pass 2026-10-21 — 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: pass 2026-10-21
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 2026-10-22 with SymPy 1.14.0.