Eliminating the parameter
Problem 8.344 · easy
Eliminate the parameter from \( \displaystyle x = t - 2,\ y = t^{2} \), and say what kind of curve it is.
- t = x − -2.
- \[ \left. - x^{2} - 4 x + y - 4 \right|_{\substack{ x=t - 2\\ y=t^{2} }} = 0 \]The parametric point satisfies the equation for every t.✓ Proved
Answer \( y = x^{2} + 4 x + 4\ (\text{parabola}) \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 first sentence incorrectly solves for t. From x = t - 2 one gets t = x + 2, not "x − -2". This mis‑statement misleads the reader about the elimination step.qwen3.6:27b-mlx: fail (error) — The solution contains a syntax error in the sentence 't = x - -2', which is ambiguous and likely intended to be 't = x + 2'. Furthermore, the equation provided in step 2 is incorrect; substituting x = t - 2 and y = t^2 into y = x^2 + 4x + 4 yields 0, but the expression shown is '-x^2 - 4x + y - 4', which equals -(x^2 + 4x - y + 4). While -(0) is 0, the standard form derived from y = (x+2)^2 is y - x^2 - 4x - 4 = 0, so the signs on the x^2 and 4x terms in the provided expression are flipped relative to the standard expansion, or rather, the expression -x^2 - 4x + y - 4 corresponds to y = x^2 + 4x + 4, so it is actually correct algebraically, but the sentence 't = x - -2' is poor style/confusing. Wait, let's re-read carefully. The equation is -x^2 - 4x + y - 4 = 0 => y = x^2 + 4x + 4. This is correct. The sentence 't = x - -2' is technically 't = x + 2' if interpreted as double negative, but written as 'x - -2' is very poor style. However, is it an error? It's ambiguous. Let's look at the substitution. x = t - 2 => t = x + 2. The sentence says 't = x - -2'. This is likely a typo for 't = x + 2'. Is it an error? It's misleading/poor style. But wait, look at the equation again. The equation is `Subs(-x**2 - 4*x + y - 4, ...) = 0`. This implies the curve is `-x^2 - 4x + y - 4 = 0`, which is `y = x^2 + 4x + 4`. This matches the stated answer. The main issue is the sentence `t = x - -2`. It is confusing. Is it an error? No, it's just bad notation. But wait, did the user make a mistake in the derivation? x = t - 2 => t = x + 2. If the student wrote t = x - -2, they might mean t = x - (-2) = x + 2. This is correct. However, usually, we substitute t into y. y = t^2 => y = (x+2)^2 => y = x^2 + 4x + 4. The solution skips the substitution step and jumps to checking the final equation. The sentence 't = x - -2' is the only sentence. It is syntactically weird. Let's call it style/misleading. But is there a deeper error? No. The verdict should probably be pass if we are lenient on ' - - ', but ' - - ' is not standard. Let's look at the prompt requirements: 'nothing would teach a student something false'. Writing 'x - -2' teaches bad notation. I will mark it as style/misleading. Actually, looking at the equation line, it is marked 'proved'. The sentence is 't = x - -2'. This is likely a typo for 't = x + 2'. I will mark it as style.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution contains a syntax error in the sentence 't = x - -2', which is ambiguous and likely intended to be 't = x + 2'. Furthermore, the equation provided in step 2 is incorrect; substituting x = t - 2 and y = t^2 into y = x^2 + 4x + 4 yields 0, but the expression shown is '-x^2 - 4x + y - 4', which equals -(x^2 + 4x - y + 4). While -(0) is 0, the standard form derived from y = (x+2)^2 is y - x^2 - 4x - 4 = 0, so the signs on the x^2 and 4x terms in the provided expression are flipped relative to the standard expansion, or rather, the expression -x^2 - 4x + y - 4 corresponds to y = x^2 + 4x + 4, so it is actually correct algebraically, but the sentence 't = x - -2' is poor style/confusing. Wait, let's re-read carefully. The equation is -x^2 - 4x + y - 4 = 0 => y = x^2 + 4x + 4. This is correct. The sentence 't = x - -2' is technically 't = x + 2' if interpreted as double negative, but written as 'x - -2' is very poor style. However, is it an error? It's ambiguous. Let's look at the substitution. x = t - 2 => t = x + 2. The sentence says 't = x - -2'. This is likely a typo for 't = x + 2'. Is it an error? It's misleading/poor style. But wait, look at the equation again. The equation is `Subs(-x**2 - 4*x + y - 4, ...) = 0`. This implies the curve is `-x^2 - 4x + y - 4 = 0`, which is `y = x^2 + 4x + 4`. This matches the stated answer. The main issue is the sentence `t = x - -2`. It is confusing. Is it an error? No, it's just bad notation. But wait, did the user make a mistake in the derivation? x = t - 2 => t = x + 2. If the student wrote t = x - -2, they might mean t = x - (-2) = x + 2. This is correct. However, usually, we substitute t into y. y = t^2 => y = (x+2)^2 => y = x^2 + 4x + 4. The solution skips the substitution step and jumps to checking the final equation. The sentence 't = x - -2' is the only sentence. It is syntactically weird. Let's call it style/misleading. But is there a deeper error? No. The verdict should probably be pass if we are lenient on ' - - ', but ' - - ' is not standard. Let's look at the prompt requirements: 'nothing would teach a student something false'. Writing 'x - -2' teaches bad notation. I will mark it as style/misleading. Actually, looking at the equation line, it is marked 'proved'. The sentence is 't = x - -2'. This is likely a typo for 't = x + 2'. I will mark it as style.gpt-oss:20b: fail (error) 2026-10-18 — The first sentence incorrectly solves for t. From x = t - 2 one gets t = x + 2, not "x − -2". This mis‑statement misleads the reader about the elimination step.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The first sentence contains a syntax error ('x - -2') and fails to explicitly solve for t (t = x + 2), which is the necessary step to substitute into the y-equation. The solution jumps from a malformed expression to a verified result without showing the derivation.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly solves for t. From x = t – 2 one should write t = x + 2, not "t = x − -2."qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution contains a syntax error in the first sentence ('t = x - -2') and the unchecked equation in line 2 represents y = x^2 + 4x + 4, but the derivation is not shown, making the 'sentence' step invalid as a logical bridge.gpt-oss:20b: fail (error) 2026-10-14 — The sentence "t = x − -2" incorrectly solves for t; it should be t = x + 2. The solution also does not properly eliminate the parameter to obtain y = x^2 + 4x + 4, so the reasoning is incomplete and incorrect.
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.