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Eliminating the parameter

Problem 8.370 · easy

Eliminate the parameter from \( \displaystyle x = 3 t + 1,\ y = - 2 t \), and say what kind of curve it is.

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

Answer \( y = \frac{2}{3} - \frac{2 x}{3}\ (\text{line}) \)

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: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The derived equation y = 2/3 - 2x/3 is incorrect; substituting x = 3t + 1 yields y = -2t = -2/3(x - 1) = -2x/3 + 2/3, but the sign of the constant te

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The derived equation y = 2/3 - 2x/3 is incorrect; substituting x = 3t + 1 yields y = -2t = -2/3(x - 1) = -2x/3 + 2/3, but the sign of the constant te
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong sign for the slope. Substituting t = (x-1)/3 into y = -2t yields y = -2(x-1)/3 = -2x/3 + 2/3, but the stated answer
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong sign for the slope. Eliminating t yields y = -2/3(x - 1) = -2x/3 + 2/3, but the stated answer is y = 2/3 - 2x/3 (whi
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.