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Level curves and domains

Problem 10.659 · easy Video

Find the equation of the level curve of \( \displaystyle f(x, y) = - 2 x^{2} - 3 y^{2} + 7 \) that passes through \( \displaystyle (1, 1) \), and describe it.

A level curve is f(x, y) = c; the point fixes c.
\[ \left. - 2 x^{2} - 3 y^{2} + 7 \right|_{\substack{ x=1\\ y=1 }} = 2 \]
c = f(1, 1).✓ Proved
So the level curve is Eq(-2*x**2 - 3*y**2 + 7, 2), a ellipse.

Answer \( - 2 x^{2} - 3 y^{2} + 7 = 2\ (\text{a ellipse}) \)

A contour map of f(x, y) = -2x² - 3y² + 7: each curve joins points where f has the same value. The level curve through (1, 1) is highlighted.

Lines: 1 proved, 2 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: fail (style) — The grammar 'a ellipse' is incorrect; it should be 'an ellipse'. While the math is correct, the sentence contains a grammatical error.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — The grammar 'a ellipse' is incorrect; it should be 'an ellipse'. While the math is correct, the sentence contains a grammatical error.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the constant c by evaluating the function at the given point and correctly identifies the resulting quadratic equation as an ellipse.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — The phrase 'a ellipse' is grammatically incorrect; it should be 'an ellipse'. While the math is correct, the grammar error is a style issue.
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/level_curves, checked 2026-10-18 with SymPy 1.14.0.