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Eliminating the parameter

Problem 8.360 · easy

Eliminate the parameter from \( \displaystyle x = t - 1,\ y = t^{2} - 1 \), and say what kind of curve it is.

t = x − -1.
\[ \left. - x^{2} - 2 x + y \right|_{\substack{ x=t - 1\\ y=t^{2} - 1 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y = x^{2} + 2 x\ (\text{parabola}) \)

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: pass
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.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — 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.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The sentence 't = x - -1' is syntactically incorrect and confusing; it should be 't = x + 1'. While the algebraic substitution in step 2 is correct, the explicit instruction for eliminating the parameter is presented with a double negative that obscures the logic.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The first sentence contains a syntax error ('x - -1') instead of the correct algebraic step 't = x + 1'. Furthermore, the solution fails to explicitly state the final Cartesian equation y = x^2 + 2x, which is required by the problem statement.
gpt-oss:20b: fail (error) 2026-10-15 — The sentence 1 incorrectly solves for t: from x = t - 1 one gets t = x + 1, not "t = x − -1". The algebraic elimination is correct, but the setup sentence is wrong.

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.