Lines and slopes
Problem 13.299 · easy
Find the equation of the line through \( \displaystyle (-2, 3) \) that is parallel to \( \displaystyle y = - \frac{x}{2} - 1 \), in slope-intercept form.
- Parallel lines share a slope. The slope is -1/2.
- \[ 3 - \frac{x + 2}{2} = 2 - \frac{x}{2} \]Point-slope form, then simplify.✓ Proved
- \[ \left. 2 - \frac{x}{2} \right|_{\substack{ x=-2 }} = 3 \]The line passes through (-2, 3).✓ Proved
Answer \( y = 2 - \frac{x}{2} \)
Lines: 2 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 |
| 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 second sentence attempts to use point‑slope form but ends up writing an identity (3-(x+2)/2 = 2-x/2) rather than the equation of the line. It does not correctly derive y = -x/2 + 2, so the setup is incorrect.qwen3.6:27b-mlx: pass — The solution correctly identifies the slope and verifies that the resulting equation satisfies the given point. The logic is sound and the final answer is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the slope and verifies that the resulting equation satisfies the given point. The logic is sound and the final answer is correct.gpt-oss:20b: fail (error) 2026-10-18 — The second sentence attempts to use point‑slope form but ends up writing an identity (3-(x+2)/2 = 2-x/2) rather than the equation of the line. It does not correctly derive y = -x/2 + 2, so the setup is incorrect.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the slope of the parallel line and verifies that the resulting equation satisfies the given point. The logic is sound and the final answer is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution contains a logical error in step 2: it presents an identity (3 - (x + 2)/2 = 2 - x/2) as if it were a calculation step, but this is just an algebraic verification of the final answer, not a derivation. Furthermore, step 3 is a verification check, not part of the derivation process. The solution fails to actually derive the equation from the point-slope form; it essentially asserts the answer and then checks it.gpt-oss:20b: fail (error) 2026-10-15 — The solution misapplies point‑slope form: it sets 3-(x+2)/2 equal to the given line 2-x/2 instead of to the new line’s y‑value. Consequently the derived equation is incorrect and the check in step 3 merely verifies the original line, not the desired one.
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 2026-10-18 with SymPy 1.14.0.