Level curves and domains
Problem 10.751 · easy Video
Find the equation of the level curve of \( \displaystyle f(x, y) = 3 x + 3 y \) that passes through \( \displaystyle (1, 2) \), and describe it.
- A level curve is f(x, y) = c; the point fixes c.Reviewed
- \[ \left. 3 x + 3 y \right|_{\substack{ x=1\\ y=2 }} = 9 \]c = f(1, 2).✓ Proved
- So the level curve is Eq(3*x + 3*y, 9), a line.Reviewed
Answer \( 3 x + 3 y = 9\ (\text{a line}) \)
✓ Nihil obstat Lines: 1 proved, 2 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | points found numerically on the claimed curve near P give f the same value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-31gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly identifies the definition of a level curve, calculates the constant c accurately, and provides the correct equation and geometric description.gpt-oss:20b: pass 2026-10-30qwen3.6:27b-mlx: pass 2026-10-30gpt-oss:20b: pass 2026-10-30
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/level_curves, checked 2026-10-31 with SymPy 1.14.0.