Lines and slopes
Problem 13.142 · easy
Find the equation of the line through \( \displaystyle (-4, -5) \) that is parallel to \( \displaystyle y = \frac{5 x}{2} - 3 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 5/2.Reviewed
- \[ \frac{5 \left(x + 4\right)}{2} - 5 = \frac{5 x}{2} + 5 \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{5 x}{2} + 5 \right|_{\substack{ x=-4 }} = -5 \]The line passes through (-4, -5).✓ Proved
Answer \( y = \frac{5 x}{2} + 5 \)
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 given line and uses the point-slope form to derive the correct equation. The verification step confirms the result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the slope from the given line and uses the point-slope form to derive the correct equation. The verification step confirms the result.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the slope and verifies the resulting equation passes through the given point. The algebraic steps are valid and the final answer is correct.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the slope and derives the correct equation. The verification step confirms the point lies on the line.gpt-oss:20b: fail (misleading) 2026-10-14 — Sentence 2 incorrectly applies point‑slope form; it equates two expressions instead of writing y = (5/2)(x+4) – 5, which would then simplify to the correct line. The student would learn a flawed method.
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.