Lines and slopes
Problem 13.139 · easy
Find the equation of the line through \( \displaystyle (-3, -2) \) that is perpendicular to \( \displaystyle y = 1 - x \), in slope-intercept form.
- \[ - \frac{1}{-1} = 1 \]Perpendicular slopes are negative reciprocals.✓ Proved
- \[ 1 \left(x + 3\right) - 2 = x + 1 \]Point-slope form, then simplify.✓ Proved
- \[ \left. x + 1 \right|_{\substack{ x=-3 }} = -2 \]The line passes through (-3, -2).✓ Proved
Answer \( y = x + 1 \)
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 (error) — Step 2 misapplies the point‑slope formula: it omits the y‑term and incorrectly writes 1*(x+3)-2 = x+1. The correct derivation is y+2 = 1*(x+3) leading to y = x+1.qwen3.6:27b-mlx: pass — The solution correctly identifies the perpendicular slope and uses the point-slope form to derive the correct equation. The steps are logically sound and algebraically verified.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the perpendicular slope and uses the point-slope form to derive the correct equation. The steps are logically sound and algebraically verified.gpt-oss:20b: fail (error) 2026-10-17 — Step 2 misapplies the point‑slope formula: it omits the y‑term and incorrectly writes 1*(x+3)-2 = x+1. The correct derivation is y+2 = 1*(x+3) leading to y = x+1.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the negative reciprocal slope and uses the point-slope form to derive the correct equation. The steps are logically sound and algebraically verified.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the negative reciprocal slope and applies the point-slope formula to derive the correct equation. The verification step confirms the point lies on the line.gpt-oss:20b: fail (misleading) 2026-10-14 — The solution’s point‑slope step is written incorrectly: it omits the y‑term and mixes the algebraic manipulation, which could mislead a student about how to derive the equation.
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.