Lines and slopes
Problem 13.427 · easy
Find the equation of the line through \( \displaystyle (-2, -2) \) that is parallel to \( \displaystyle y = x - 3 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 1.
- \[ 1 \left(x + 2\right) - 2 = x \]Point-slope form, then simplify.✓ Proved
- \[ \left. x \right|_{\substack{ x=-2 }} = -2 \]The line passes through (-2, -2).✓ Proved
Answer \( y = x \)
Lines: 2 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 |
| 3 | ✓ 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 | the line meets the point with the required slope |
Reviewers
gpt-oss:20b: fail (error) — The solution’s equations are incorrect. Sentence 2 miswrites the point‑slope form as "1*(x + 2) - 2 = x" instead of "y + 2 = 1·(x + 2)" and then incorrectly claims the line simplifies to y = x. Sentence 3 is nonsensical; substituting x = –2 into the equation should yield y = –2, not the stated expression. The steps do not correctly derive the required line.qwen3.6:27b-mlx: fail (misleading) — The solution provides no explanation for how the final answer y = x is derived from the point-slope form, skipping the necessary algebraic simplification steps. Additionally, step 3 is a verification of a point rather than a derivation of the equation, making the logical flow disjointed.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-22 — The solution provides no explanation for how the final answer y = x is derived from the point-slope form, skipping the necessary algebraic simplification steps. Additionally, step 3 is a verification of a point rather than a derivation of the equation, making the logical flow disjointed.gpt-oss:20b: fail (error) 2026-10-22 — The solution’s equations are incorrect. Sentence 2 miswrites the point‑slope form as "1*(x + 2) - 2 = x" instead of "y + 2 = 1·(x + 2)" and then incorrectly claims the line simplifies to y = x. Sentence 3 is nonsensical; substituting x = –2 into the equation should yield y = –2, not the stated expression. The steps do not correctly derive the required line.qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution fails to explicitly state the final equation of the line in slope-intercept form as requested. While the algebraic steps are correct, the final output 'y = x' is not derived or stated as the answer to the problem.gpt-oss:20b: inconclusive 2026-10-21 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope step is incorrect: it should be \(y+2=1(x+2)\), not the expression given."}qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution fails to explicitly state the final equation in slope-intercept form as requested. Step 2 presents an identity rather than deriving the equation, and Step 3 is a verification check, not the derivation of the result.gpt-oss:20b: fail (error) 2026-10-21 — The point‑slope step is incorrectly written; it should be y+2 = 1·(x+2), not 1*(x+2) - 2 = x. This misstates the equation of the line.
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/lines_slopes, checked 2026-10-22 with SymPy 1.14.0.