Lines and slopes
Problem 13.297 · easy
Find the equation of the line through \( \displaystyle (-5, 5) \) that is parallel to \( \displaystyle y = \frac{2 x}{3} + 3 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 2/3.Reviewed
- \[ \frac{2 \left(x + 5\right)}{3} + 5 = \frac{2 x}{3} + \frac{25}{3} \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{2 x}{3} + \frac{25}{3} \right|_{\substack{ x=-5 }} = 5 \]The line passes through (-5, 5).✓ Proved
Answer \( y = \frac{2 x}{3} + \frac{25}{3} \)
✓ 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 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 and uses the point-slope form to derive the equation. The algebraic steps are verified, and the final answer is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the slope and derives the correct equation. The verification steps are algebraically sound.gpt-oss:20b: pass 2026-10-15
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.