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Eliminating the parameter

Problem 8.366 · easy

Eliminate the parameter from \( \displaystyle x = - 2 t - 2,\ y = t - 3 \), and say what kind of curve it is.

t = (x − -2)/-2; substitute into y.
Reviewed
\[ \left. \frac{x}{2} + y + 4 \right|_{\substack{ x=- 2 t - 2\\ y=t - 3 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y = - \frac{x}{2} - 4\ (\text{line}) \)

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 as a line. The algebraic steps are valid.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly eliminates the parameter and identifies the curve as a line. The algebraic steps are valid.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly eliminates the parameter and identifies the curve as a line. The algebraic steps are sound.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The algebraic manipulation in step 1 is incorrect: solving x = -2t - 2 for t yields t = (-x - 2) / 2, not (x + 2) / -2 (which simplifies to the same, but the substitution logic in step 2 is garbled). More critically, step 2 presents an equation `Subs(...) = 0` that does not clearly derive the stated answer `y = -x/2 - 4`. If we substitute t = -(x+2)/2 into y = t - 3, we get y = -(x+2)/2 - 3 = -x/2 - 1 - 3 = -x/2 - 4. The solution's step 2 is syntactically confusing and does not explicitly show the correct substitution leading to the final result, making the derivation opaque and potentially misleading about the sign handling.
gpt-oss:20b: pass 2026-10-15

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.