Lines and slopes
Problem 13.301 · easy
Find the equation of the line through \( \displaystyle (2, -5) \) that is parallel to \( \displaystyle y = - \frac{2 x}{3} - 4 \), in slope-intercept form.
- Parallel lines share a slope. The slope is -2/3.
- \[ - \frac{2 \left(x - 2\right)}{3} - 5 = - \frac{2 x}{3} - \frac{11}{3} \]Point-slope form, then simplify.✓ Proved
- \[ \left. - \frac{2 x}{3} - \frac{11}{3} \right|_{\substack{ x=2 }} = -5 \]The line passes through (2, -5).✓ Proved
Answer \( y = - \frac{2 x}{3} - \frac{11}{3} \)
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 point‑slope form is written incorrectly: the left side should be y+5, not just the expression. The equation should read y+5 = -2/3(x-2), which simplifies to y = -2/3x - 11/3.qwen3.6:27b-mlx: fail (misleading) — The solution skips the explicit derivation of the slope-intercept form, jumping from the slope to a verified equation. While the final answer is correct, the reasoning presented in step 2 is a verification, not a derivation, which fails to teach the student how to construct the equation from the point and slope.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The solution skips the explicit derivation of the slope-intercept form, jumping from the slope to a verified equation. While the final answer is correct, the reasoning presented in step 2 is a verification, not a derivation, which fails to teach the student how to construct the equation from the point and slope.gpt-oss:20b: fail (error) 2026-10-18 — The point‑slope form is written incorrectly: the left side should be y+5, not just the expression. The equation should read y+5 = -2/3(x-2), which simplifies to y = -2/3x - 11/3.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the slope of the parallel line and uses the point-slope form to derive the equation. The algebraic steps are verified, and the final result matches the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the slope of the parallel line and uses the point-slope form to derive the equation. The final answer is correct and in the requested slope-intercept form.gpt-oss:20b: fail (error) 2026-10-15 — The point‑slope step is incorrectly written: it omits the y‑term and misapplies the formula. The correct form is y+5 = (-2/3)(x-2).
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.