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Level curves and domains

Problem 10.746 · easy Video

Find the equation of the level curve of \( \displaystyle f(x, y) = 3 x + y \) that passes through \( \displaystyle (-1, 3) \), and describe it.

A level curve is f(x, y) = c; the point fixes c.
Reviewed
\[ \left. 3 x + y \right|_{\substack{ x=-1\\ y=3 }} = 0 \]
c = f(-1, 3).✓ Proved
So the level curve is Eq(3*x + y, 0), a line.
Reviewed

Answer \( 3 x + y = 0\ (\text{a line}) \)

A contour map of f(x, y) = 3x + y: each curve joins points where f has the same value. The level curve through (-1, 3) is highlighted.

Lines: 1 proved, 2 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the level curve equation and its geometric description. The logic is sound and the steps are clear.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly identifies the level curve equation and its geometric description. The logic is sound and the steps are clear.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly identifies the method for finding a level curve, calculates the constant c accurately, and provides the correct equation and geometric description.
gpt-oss:20b: pass 2026-10-30
qwen3.6:27b-mlx: fail (error) 2026-10-30 — The calculation of c is incorrect: f(-1, 3) = 3(-1) + 3 = 0, so the equation is 3x + y = 0. Wait, the calculation IS correct. Let me re-read carefully. f(-1,3) = -3 + 3 = 0. So c=0. The equation is 3x+y=0. The stated answer is 3x+y=0. The solution says c=0. This is actually correct. Why did I think it was wrong? Let me check the prompt again. 'Stated answer: 3 x + y = 0'. Solution step 2 says Subs... = 0. Step 3 says Eq(3*x+y, 0). This is correct. Is there a subtle error? 'describe it'. It says 'a line'. That is correct. Is the format of the sentence wrong? No. Is the math wrong? No. Wait, usually these adversarial reviews find an error. Let me look really closely. f(x,y) = 3x+y. Point (-1,3). 3(-1)+3 = 0. c=0. Level curve 3x+y=0. This is a line. Everything seems correct. Did I miss something? Maybe the 'unchecked' tag on the equation implies I shouldn't check the algebra, but I should check if the sentence claims the result of the algebra correctly? The sentence says 'the point fixes c'. The equation shows c=0. The next sentence says 'So the level curve is... 0'. This is logically sound. Is it possible the problem implies a non-zero constant? No. Is it possible the description 'a line' is insufficient? No, it's a linear function, so level curves are lines. Let me reconsider if there is a trick. Maybe the point is (-1, 3) but the function is different? No, f(x,y)=3x+y. Actually, looking at similar examples in this dataset style, often the 'error' is a sign error or arithmetic error. Here 3(-1)+3=0. It is correct. Is it possible the verdict should be pass? The instructions say 'Say pass with empty notes if it is sound.' It is sound. Wait, let me look at the JSON structure required. Let me double check the arithmetic one more time. 3 * -1 = -3. -3 + 3 = 0. Correct. Is there any ambiguity in 'describe it'? 'A line' is a correct description. I will mark it as pass.
gpt-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.