Eliminating the parameter
Problem 8.364 · medium
Eliminate the parameter from \( \displaystyle x = e^{t},\ y = e^{t} \), and say what kind of curve it is.
- x = eᵗ, so y = e^(1t) = (eᵗ)^1 (with x > 0).
- \[ \left. - x + y \right|_{\substack{ x=e^{t}\\ y=e^{t} }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = x\ (\text{power curve}) \)
Lines: 1 proved, 1 not checked. 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 | 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: fail (error) — The solution incorrectly states that the curve is a power curve; in fact the eliminated equation is y=x, which is a straight line. The first sentence is unnecessary and does not contribute to elimination.qwen3.6:27b-mlx: fail (misleading) — The solution fails to eliminate the parameter to produce an explicit Cartesian equation (y = x). Instead, it merely verifies that the parametric equations satisfy a relation, which does not constitute elimination. Additionally, labeling y=x as a 'power curve' is non-standard and potentially confusing terminology.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution fails to eliminate the parameter to produce an explicit Cartesian equation (y = x). Instead, it merely verifies that the parametric equations satisfy a relation, which does not constitute elimination. Additionally, labeling y=x as a 'power curve' is non-standard and potentially confusing terminology.gpt-oss:20b: fail (error) 2026-10-18 — The solution incorrectly states that the curve is a power curve; in fact the eliminated equation is y=x, which is a straight line. The first sentence is unnecessary and does not contribute to elimination.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The solution fails to eliminate the parameter to find the relationship between x and y (y=x). It merely verifies that the parametric equations satisfy a tautology. Furthermore, it incorrectly classifies the curve as a 'power curve' without noting the domain restriction x > 0, which distinguishes it from the full line y=x.gpt-oss:20b: pass 2026-10-16 — The solution correctly eliminates the parameter and identifies the curve as y=x. The first sentence is superfluous and could be clearer, but it does not introduce a mathematical error.qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution fails to eliminate the parameter to find the Cartesian equation y = x. Instead, it merely verifies that the parametric equations satisfy an undefined linear equation. Furthermore, it misidentifies the curve as a 'power curve' (typically y = x^n) rather than a line.gpt-oss:20b: fail (error) 2026-10-15 — The solution does not eliminate the parameter or derive the relation y=x. Sentence 1 is trivial and does not provide the required curve equation, and the subsequent equation is merely a tautology. The answer is incomplete and incorrect.
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 2026-10-18 with SymPy 1.14.0.