Eliminating the parameter
Problem 8.538 · easy
Eliminate the parameter from \( \displaystyle x = t - 1,\ y = t^{2} - 2 \), and say what kind of curve it is.
- t = x − -1.
- \[ \left. - x^{2} - 2 x + y + 1 \right|_{\substack{ x=t - 1\\ y=t^{2} - 2 }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = x^{2} + 2 x - 1\ (\text{parabola}) \)
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) — Sentence 1 incorrectly solves for t: it should be t = x + 1, not t = x − -1. This mis‑states the relationship between the parameter and x, leading to an incorrect elimination step.qwen3.6:27b-mlx: fail (error) — The first sentence contains a syntax error ('x - -1') instead of the correct algebraic step 't = x + 1'. This makes the derivation unclear and technically incorrect as written.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-04 — The first sentence contains a syntax error ('x - -1') instead of the correct algebraic step 't = x + 1'. This makes the derivation unclear and technically incorrect as written.gpt-oss:20b: fail (error) 2026-11-04 — Sentence 1 incorrectly solves for t: it should be t = x + 1, not t = x − -1. This mis‑states the relationship between the parameter and x, leading to an incorrect elimination step.qwen3.6:27b-mlx: fail (error) 2026-11-03 — The first sentence contains a syntax error ('x - -1') and fails to explicitly solve for t (t = x + 1), which is the necessary step to eliminate the parameter. The solution jumps to a verification step without showing the derivation.gpt-oss:20b: fail (error) 2026-11-03 — The first sentence incorrectly solves for t: from x = t – 1 one should have t = x + 1, not “x – -1.” This mis‑statement propagates through the elimination and the final result is not justified. The solution also omits a clear statement that the resulting equation represents a parabola.qwen3.6:27b-mlx: fail (error) 2026-11-03 — The sentence in step 1 contains a syntax error ('x - -1') and fails to explicitly solve for t (it should state t = x + 1). Furthermore, the solution skips the crucial step of substituting t into the equation for y to derive the Cartesian equation, jumping straight to a verification step.gpt-oss:20b: pass 2026-11-03
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 2026-11-04 with SymPy 1.14.0.