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Eliminating the parameter

Problem 8.342 · easy

Eliminate the parameter from \( \displaystyle x = 2 - 2 t,\ y = 2 - 4 t \), and say what kind of curve it is.

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

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

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 verification confirms the relationship.

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 verification confirms the relationship.
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 verification confirms the relationship.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The derived equation y = 2x - 2 is incorrect; substituting x = 2 - 2t into y = 2 - 4t yields y = 2x, not y = 2x - 2. The constant term is wrong.
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.