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Eliminating the parameter

Problem 8.338 · easy

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

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

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

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly eliminates the parameter and identifies the curve. The algebraic verification in line 2 confirms the derived Cartesian equation is satisfied by the parametric equations.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly eliminates the parameter and identifies the curve. The algebraic verification in line 2 confirms the derived Cartesian equation is satisfied by the parametric equations.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly state the final Cartesian equation y = x^2 - 2x + 2, which is required by the problem statement. It only verifies that the parametric equations satisfy an implicit relation without deriving or stating the explicit function y(x).
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly eliminates the parameter and identifies the curve as a parabola. The algebraic verification in step 2 confirms the derived Cartesian equation.
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 2026-10-18 with SymPy 1.14.0.