Lines and slopes
Problem 13.505 · easy
Find the equation of the line through \( \displaystyle (1, 4) \) that is perpendicular to \( \displaystyle y = 2 x - 3 \), in slope-intercept form.
- Perpendicular slopes are negative reciprocals: -1/2.
- \[ 4 - \frac{x - 1}{2} = \frac{9}{2} - \frac{x}{2} \]Point-slope form, then simplify.✓ Proved
- \[ \left. - \frac{x}{2} + \frac{9}{2} \right|_{\substack{ x=1 }} = 4 \]The line passes through (1, 4).✓ Proved
Answer \( y = \frac{9}{2} - \frac{x}{2} \)
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 solution’s second line misapplies point‑slope form. It writes 4-(x-1)/2=9/2-x/2, which is not equivalent to y-4=-½(x-1); the algebra is incorrect and the resulting equation does not represent the desired line.qwen3.6:27b-mlx: pass — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The verification step confirms the point lies on the line.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly identifies the negative reciprocal slope and derives the correct equation. The verification step confirms the point lies on the line.gpt-oss:20b: fail (error) 2026-11-13 — The solution’s second line misapplies point‑slope form. It writes 4-(x-1)/2=9/2-x/2, which is not equivalent to y-4=-½(x-1); the algebra is incorrect and the resulting equation does not represent the desired line.qwen3.6:27b-mlx: fail (error) 2026-11-13 — The solution fails to explicitly state the point-slope equation (y - 4 = -1/2(x - 1)) before simplifying. Line 2 presents an algebraic identity that is true but does not clearly demonstrate the derivation of the final slope-intercept form from the point-slope form, making the logical step opaque.gpt-oss:20b: pass 2026-11-13
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 2026-11-13 with SymPy 1.14.0.