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Eliminating the parameter

Problem 8.345 · easy

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

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

Answer $ y = \frac{x}{3} - 3\ (\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 sentence claims t = x/3, but the equation substitutes -x/3, which is algebraically incorrect. The correct substitution yields y = x/3 - 3.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The sentence claims t = x/3, but the equation substitutes -x/3, which is algebraically incorrect. The correct substitution yields y = x/3 - 3.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The sentence claims t = x/3, but the equation substitutes -x/3, which is algebraically incorrect. The correct substitution should yield y = x/3 - 3, not y = -x/3 - 3.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the parameter $t$ in terms of $x$ and substitutes it into the equation for $y$ to obtain the Cartesian equation. The resulting equation represents a line, which is the correct classification.
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.