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Eliminating the parameter

Problem 8.234 · easy

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

t = (x − -1)/-2; substitute into y.
\[ \left. \frac{\left(-1\right) x}{2} + y + 1 \cdot \frac{1}{2} \right|_{\substack{ x=- 2 t - 1\\ y=- t - 1 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y = \frac{x}{2} - \frac{1}{2}\ (\text{line}) \)

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 solution fails to state the final Cartesian equation explicitly. While line 2 confirms the parametric point satisfies an implicit relation, it does not provide the explicit form y = x/2 - 1/2 requested by the problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to state the final Cartesian equation explicitly. While line 2 confirms the parametric point satisfies an implicit relation, it does not provide the explicit form y = x/2 - 1/2 requested by the problem statement.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly eliminates the parameter and identifies the curve as a line. The algebraic steps are sound.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly eliminates the parameter and identifies the curve as a line. The algebraic steps are sound.
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 2026-10-17 with SymPy 1.14.0.