Lines and slopes
Problem 13.137 · easy
Find the equation of the line through \( \displaystyle (-2, -5) \) that is parallel to \( \displaystyle y = \frac{5 x}{3} + 4 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 5/3.
- \[ \frac{5 \left(x + 2\right)}{3} - 5 = \frac{5 x}{3} - \frac{5}{3} \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{5 x}{3} - \frac{5}{3} \right|_{\substack{ x=-2 }} = -5 \]The line passes through (-2, -5).✓ Proved
Answer \( y = \frac{5 x}{3} - \frac{5}{3} \)
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: passqwen3.6:27b-mlx: fail (style) — The solution skips the explicit derivation of the point-slope form equation y - y1 = m(x - x1), jumping directly to a simplified algebraic identity. While the final answer is correct, the lack of intermediate steps makes it difficult for a student to follow the logical connection between the given point, the slope, and the final equation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-17 — The solution skips the explicit derivation of the point-slope form equation y - y1 = m(x - x1), jumping directly to a simplified algebraic identity. While the final answer is correct, the lack of intermediate steps makes it difficult for a student to follow the logical connection between the given point, the slope, and the final equation.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the slope from the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the result.gpt-oss:20b: fail (error) 2026-10-15 — The solution’s second line claims to use point‑slope form but omits the variable y and incorrectly sets the expression 5*(x+2)/3‑5 equal to 5x/3‑5/3. This does not represent the line equation and misleads the reader.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims to use point-slope form but writes an equation that is not in point-slope form (y - y1 = m(x - x1)). It presents a simplified algebraic identity as the derivation step, skipping the actual setup and making the logic opaque.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.