Lines and slopes
Problem 13.345 · easy
Find the equation of the line through \( \displaystyle (-5, -4) \) that is parallel to \( \displaystyle y = \frac{3 x}{2} + 1 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 3/2.
- \[ \frac{3 \left(x + 5\right)}{2} - 4 = \frac{3 x}{2} + \frac{7}{2} \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{3 x}{2} + \frac{7}{2} \right|_{\substack{ x=-5 }} = -4 \]The line passes through (-5, -4).✓ Proved
Answer \( y = \frac{3 x}{2} + \frac{7}{2} \)
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 point‑slope form is applied incorrectly: it should be y-(-4)= frac32(x-(-5)), not the expression shown.qwen3.6:27b-mlx: pass — The solution correctly identifies the slope and uses the point-slope form to derive the equation. The algebraic steps are verified, and the final answer is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the slope and uses the point-slope form to derive the equation. The algebraic steps are verified, and the final answer is correct.gpt-oss:20b: fail (error) 2026-10-19 — The point‑slope form is applied incorrectly: it should be y-(-4)= frac32(x-(-5)), not the expression shown.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 correct equation. The verification step confirms the point lies on the line.gpt-oss:20b: fail (misleading) 2026-10-16 — The sentence 2 claims to use point‑slope form, but the displayed equation 3*(x + 5)/2 - 4 = 3*x/2 + 7/2 is not the correct point‑slope expression. It misleads a student into thinking that form is valid, whereas the correct point‑slope equation is y + 4 = (3/2)(x + 5).qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims to use point-slope form but presents an equation that mixes point-slope and slope-intercept structures incorrectly. The standard point-slope form is y - y1 = m(x - x1), which would be y - (-4) = 3/2(x - (-5)). The provided equation 3*(x + 5)/2 - 4 = 3*x/2 + 7/2 is not a standard derivation step and obscures the logic.gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.