Eliminating the parameter
Problem 8.485 · easy
Eliminate the parameter from \( \displaystyle x = t - 1,\ y = t^{2} + 3 \), and say what kind of curve it is.
- t = x − -1.
- \[ \left. - x^{2} - 2 x + y - 4 \right|_{\substack{ x=t - 1\\ y=t^{2} + 3 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = x^{2} + 2 x + 4\ (\text{parabola}) \)
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 first sentence incorrectly solves for t; from x = t – 1 we have t = x + 1, not t = x – –1. The subsequent equation is correct, but the erroneous sentence misrepresents the algebraic manipulation.qwen3.6:27b-mlx: fail (error) — The sentence 't = x - -1' contains a syntax error and is mathematically ambiguous; it should be 't = x + 1'. While the final equation is correct, the intermediate step is not a valid mathematical statement.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The sentence 't = x - -1' contains a syntax error and is mathematically ambiguous; it should be 't = x + 1'. While the final equation is correct, the intermediate step is not a valid mathematical statement.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly solves for t; from x = t – 1 we have t = x + 1, not t = x – –1. The subsequent equation is correct, but the erroneous sentence misrepresents the algebraic manipulation.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The first sentence contains a syntax error ('x - -1') instead of the correct substitution 't = x + 1'. While the final equation is correct, the intermediate step is mathematically ill-formed and misleading.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly solves for t. From x = t – 1 we have t = x + 1, not "t = x − -1". This mis‑statement misleads the reader about the algebraic manipulation.qwen3.6:27b-mlx: fail (error) 2026-10-18 — The first sentence contains a syntax error ('x - -1') and fails to explicitly solve for t (should be t = x + 1). The second line is an unchecked algebraic verification rather than a derivation, and the intermediate step required to eliminate the parameter is missing.gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.