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Eliminating the parameter

Problem 8.368 · easy

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

t = (x − 3)/-1; substitute into y.
\[ \left. 2 x + y - 6 \right|_{\substack{ x=3 - t\\ y=2 t }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y = 6 - 2 x\ (\text{line}) \)

Lines: 1 proved, 1 not checked. 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	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 (style) — [domain objection, downgraded to style] The solution claims the curve is a line, but fails to specify that it is a ray (x <= 3) because the parameter t is unrestricted in the standard context unless specified, but typically 'eliminate the parameter' implies finding the Cartesian equation. However, the bigger issue is that the solution does not explicitly state the domain restriction implied by the parameter if t were restricted, but more importantly, the prompt asks 'what kind of curve'. A line is correct for the infinite parameter range. Wait, let's look closer. The solution says 't = (x-3)/-1'. This is t = 3-x. Then y = 2(3-x) = 6-2x. This is correct algebra. Is there a restriction? If t is any real number, x is any real number, so it is the entire line. The answer 'line' is correct. Why would this fail? Let's re-read the adversarial instructions. 'Check that... the setup models the problem as stated'. Problem: Eliminate parameter... say what kind of curve. Solution: 1. t = (x-3)/-1. 2. Equation proved. It misses the explicit conclusion sentence stating the final equation y = 6 - 2x and identifying it as a line. The 'Stated answer' is provided separately, but the 'Solution' block only contains the derivation steps. Step 1 is a sentence. Step 2 is an equation. The solution does not explicitly state the final Cartesian equation or the type of curve in the solution text itself, relying on the 'Stated answer' field. Usually, the solution should contain the final assertion. However, often in these formats, the 'Stated answer' is the target and the solution is the work. If the work is correct, it passes. Let's look for a subtle error. x = 3 - t => t = 3 - x. y = 2t => y = 2(3-x) = 6 - 2x. This is a line. Is there a domain issue? If t is not restricted, x ranges over all reals. So it is the entire line. Maybe the error is in the sentence 't = (x - 3)/-1'. This is algebraically correct but stylistically poor? No, severity 'style' is for 'correct, but a sentence could be clearer'. Let's look at the 'Stated answer': y = 6 - 2x (line). The solution steps: 1. t = (x-3)/-1; substitute into y. 2. Equation check. The solution does not actually *perform* the substitution in the text to show y = 6 - 2x. It just says 'substitute into y' and then checks that the point satisfies 2x+y-6=0. It doesn't explicitly derive y = 6 - 2x. A student might not know how to get from 'substitute' to the final form without seeing it. But step 2 proves the relationship. Actually, looking at similar problems, usually 'eliminate the parameter' requires writing the final equation. The solution block is incomplete because it doesn't state the final equation y = 6 - 2x. It only verifies a relation. Also, 'what kind of curve it is'. The solution does not answer this part in the text. Therefore, the solution is incomplete/misleading because it doesn't explicitly state the final result derived.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution claims the curve is a line, but fails to specify that it is a ray (x <= 3) because the parameter t is unrestricted in the standard context unless specified, but typically 'eliminate the parameter' implies finding the Cartesian equation. However, the bigger issue is that the solution does not explicitly state the domain restriction implied by the parameter if t were restricted, but more importantly, the prompt asks 'what kind of curve'. A line is correct for the infinite parameter range. Wait, let's look closer. The solution says 't = (x-3)/-1'. This is t = 3-x. Then y = 2(3-x) = 6-2x. This is correct algebra. Is there a restriction? If t is any real number, x is any real number, so it is the entire line. The answer 'line' is correct. Why would this fail? Let's re-read the adversarial instructions. 'Check that... the setup models the problem as stated'. Problem: Eliminate parameter... say what kind of curve. Solution: 1. t = (x-3)/-1. 2. Equation proved. It misses the explicit conclusion sentence stating the final equation y = 6 - 2x and identifying it as a line. The 'Stated answer' is provided separately, but the 'Solution' block only contains the derivation steps. Step 1 is a sentence. Step 2 is an equation. The solution does not explicitly state the final Cartesian equation or the type of curve in the solution text itself, relying on the 'Stated answer' field. Usually, the solution should contain the final assertion. However, often in these formats, the 'Stated answer' is the target and the solution is the work. If the work is correct, it passes. Let's look for a subtle error. x = 3 - t => t = 3 - x. y = 2t => y = 2(3-x) = 6 - 2x. This is a line. Is there a domain issue? If t is not restricted, x ranges over all reals. So it is the entire line. Maybe the error is in the sentence 't = (x - 3)/-1'. This is algebraically correct but stylistically poor? No, severity 'style' is for 'correct, but a sentence could be clearer'. Let's look at the 'Stated answer': y = 6 - 2x (line). The solution steps: 1. t = (x-3)/-1; substitute into y. 2. Equation check. The solution does not actually *perform* the substitution in the text to show y = 6 - 2x. It just says 'substitute into y' and then checks that the point satisfies 2x+y-6=0. It doesn't explicitly derive y = 6 - 2x. A student might not know how to get from 'substitute' to the final form without seeing it. But step 2 proves the relationship. Actually, looking at similar problems, usually 'eliminate the parameter' requires writing the final equation. The solution block is incomplete because it doesn't state the final equation y = 6 - 2x. It only verifies a relation. Also, 'what kind of curve it is'. The solution does not answer this part in the text. Therefore, the solution is incomplete/misleading because it doesn't explicitly state the final result derived.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly eliminates the parameter to find the Cartesian equation and correctly identifies the curve as a line.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly eliminates the parameter and identifies the curve as a line. The algebraic verification in step 2 is consistent with the result.
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.