Lines and slopes
Problem 13.123 · easy
Find the equation of the line through \( \displaystyle (0, -2) \) that is perpendicular to \( \displaystyle y = - \frac{x}{3} - 3 \), in slope-intercept form.
- \[ - \frac{1}{\left(-1\right) \frac{1}{3}} = 3 \]Perpendicular slopes are negative reciprocals.✓ Proved
- \[ 3 \left(x + 0\right) - 2 = 3 x - 2 \]Point-slope form, then simplify.✓ Proved
- \[ \left. 3 x - 2 \right|_{\substack{ x=0 }} = -2 \]The line passes through (0, -2).✓ Proved
Answer \( y = 3 x - 2 \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the negative reciprocal slope and applies the point-slope formula accurately. The steps are logically sound and lead to the correct result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the negative reciprocal slope and applies the point-slope formula accurately. The steps are logically sound and lead to the correct result.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the negative reciprocal slope and applies the point-slope formula accurately to derive the final equation.gpt-oss:20b: fail (error) 2026-10-14 — Step 2 incorrectly applies point‑slope form; it omits the y‑term and therefore does not correctly represent the line. The correct point‑slope form is y+2=3(x-0), which simplifies to y=3x-2.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the negative reciprocal slope and applies the point-slope formula accurately to derive the final equation.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.