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Eliminating the parameter

Problem 8.223 · medium

Eliminate the parameter from \( \displaystyle x = e^{t},\ y = e^{4 t} \), and say what kind of curve it is.

x = eᵗ, so y = e^(4t) = (eᵗ)^4 (with x > 0).
\[ \left. - x^{4} + y \right|_{\substack{ x=e^{t}\\ y=e^{4 t} }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y = x^{4}\ (\text{power curve}) \)

Lines: 1 proved, 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	points on the parametric curve at four values of t satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to specify the domain restriction x > 0, which is required because x = e^t is always positive. Without this restriction, the Cartesian equation y = x^4 describes a curve defined for all real x, whereas the parametric curve only covers x > 0.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution fails to specify the domain restriction x > 0, which is required because x = e^t is always positive. Without this restriction, the Cartesian equation y = x^4 describes a curve defined for all real x, whereas the parametric curve only covers x > 0.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution fails to specify the domain restriction x > 0, which is required because x = e^t is always positive. Without this restriction, the Cartesian equation y = x^4 describes a curve that includes negative x values, which are not part of the original parametric path.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution fails to specify the domain restriction x > 0, which is required because x = e^t is always positive. Without this restriction, y = x^4 represents a different curve (defined for all real x) than the parametric one.
gpt-oss:20b: pass 2026-10-14

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/parametric_eliminate, checked 2026-10-17 with SymPy 1.14.0.