Lines and slopes
Problem 13.422 · easy
Find the equation of the line through \( \displaystyle (-2, -2) \) that is parallel to \( \displaystyle y = - \frac{4 x}{3} - 2 \), in slope-intercept form.
- Parallel lines share a slope. The slope is -4/3.
- \[ - \frac{4 \left(x + 2\right)}{3} - 2 = - \frac{4 x}{3} - \frac{14}{3} \]Point-slope form, then simplify.✓ Proved
- \[ \left. - \frac{4 x}{3} - \frac{14}{3} \right|_{\substack{ x=-2 }} = -2 \]The line passes through (-2, -2).✓ Proved
Answer \( y = - \frac{4 x}{3} - \frac{14}{3} \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope step is incorrect: it omits the y‑term and misapplies the formula. The correct equation should be \(y+2=-\frac{4}{3}(x+2)\), which simplifqwen3.6:27b-mlx: pass — The solution correctly identifies the slope of the parallel line and verifies that the resulting equation passes through the given point. The algebraic steps are correct and the final answer matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly identifies the slope of the parallel line and verifies that the resulting equation passes through the given point. The algebraic steps are correct and the final answer matches the stated answer.gpt-oss:20b: inconclusive 2026-10-22 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope step is incorrect: it omits the y‑term and misapplies the formula. The correct equation should be \(y+2=-\frac{4}{3}(x+2)\), which simplifqwen3.6:27b-mlx: fail (misleading) 2026-10-21 — The solution claims to use point-slope form but presents an equation that incorrectly sets the original line's expression equal to the result, rather than showing the derivation from y - y1 = m(x - x1). This obscures the actual method and could mislead a student about how to apply the point-slope formula.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: fail (error) 2026-10-21 — The solution claims to use point-slope form in step 2, but the equation shown is not in point-slope form (which would be y - y1 = m(x - x1)). Instead, it presents an equality between two expressions without defining the left-hand side as y, making the derivation logically incoherent.gpt-oss:20b: fail (error) 2026-10-21 — The point‑slope form is written incorrectly: it omits the variable y and misapplies the formula. The correct form is y+2 = (-4/3)(x+2), which simplifies to y = -4x/3 - 14/3. The solution’s equation line is therefore wrong.
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.