Lines and slopes
Problem 13.305 · easy
Find the equation of the line through \( \displaystyle (3, -4) \) that is perpendicular to \( \displaystyle y = - 2 x - 3 \), in slope-intercept form.
- \[ - \frac{1}{-2} = \frac{1}{2} \]Perpendicular slopes are negative reciprocals.✓ Proved
- \[ \frac{x - 3}{2} - 4 = \frac{x}{2} - \frac{11}{2} \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{x}{2} - \frac{11}{2} \right|_{\substack{ x=3 }} = -4 \]The line passes through (3, -4).✓ Proved
Answer \( y = \frac{x}{2} - \frac{11}{2} \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 (misleading) — Step 2 incorrectly applies the point‑slope formula; the left side should involve y, e.g. y+4 = (1/2)(x-3), not (x-3)/2-4. This misleads a student about how to form the equation.qwen3.6:27b-mlx: pass — The solution correctly identifies the negative reciprocal slope and verifies the point-slope form. The algebraic steps are valid and lead to the correct slope-intercept equation.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the negative reciprocal slope and verifies the point-slope form. The algebraic steps are valid and lead to the correct slope-intercept equation.gpt-oss:20b: fail (misleading) 2026-10-18 — Step 2 incorrectly applies the point‑slope formula; the left side should involve y, e.g. y+4 = (1/2)(x-3), not (x-3)/2-4. This misleads a student about how to form the equation.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the negative reciprocal slope and verifies the point-slope form. The algebraic steps are valid and lead to the correct slope-intercept equation.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly state the slope-intercept form equation derived from the point-slope form, jumping directly to a verification step. While the final answer is correct, the logical flow is broken because the derivation of the final equation is missing; line 2 is an identity check, not a derivation.gpt-oss:20b: pass 2026-10-15
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.