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Eliminating the parameter

Problem 8.347 · easy

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

t = (x − -2)/-1; substitute into y.
Reviewed
\[ \left. 3 x + y + 3 \right|_{\substack{ x=- t - 2\\ y=3 t + 3 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y = - 3 x - 3\ (\text{line}) \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 identifies the curve as a line and provides the correct Cartesian equation. The steps are logically sound and the algebra is verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the curve as a line and provides the correct Cartesian equation. The steps are logically sound and the algebra is verified.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the curve as a line and provides the correct Cartesian equation. The steps shown are logically sound and sufficient.
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 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 2026-10-18 with SymPy 1.14.0.