Home›Calculus 2›Eliminating the parameter›Problem 8.222

Eliminating the parameter

Problem 8.222 · easy

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

t = x − 0.
\[ \left. - x^{2} + y - 3 \right|_{\substack{ x=t\\ y=t^{2} + 3 }} = 0 \]
The parametric point satisfies the equation for every t.✓ Proved

Answer \( y = x^{2} + 3\ (\text{parabola}) \)

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: fail (error) — The solution does not actually eliminate the parameter or provide the resulting equation; it merely restates the trivial identity t=x and substitutes without concluding y=x^2+3. The steps are incomplete and do not demonstrate the required result.
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly eliminate the parameter to derive the Cartesian equation y = x^2 + 3. It only verifies that the parametric equations satisfy a relation, but does not show the derivation or state the final Cartesian form as requested.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution fails to explicitly eliminate the parameter to derive the Cartesian equation y = x^2 + 3. It only verifies that the parametric equations satisfy a relation, but does not show the derivation or state the final Cartesian form as requested.
gpt-oss:20b: fail (error) 2026-10-17 — The solution does not actually eliminate the parameter or provide the resulting equation; it merely restates the trivial identity t=x and substitutes without concluding y=x^2+3. The steps are incomplete and do not demonstrate the required result.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly state the final Cartesian equation y = x^2 + 3, which is the primary requirement of the problem. It only verifies that the parametric equations satisfy an implicit relation without deriving or stating the explicit function requested.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly state the final Cartesian equation y = x^2 + 3, which is the primary requirement of the problem. It only verifies that the parametric equations satisfy an implicit form without solving for y in terms of x.
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 2026-10-17 with SymPy 1.14.0.