Lines and slopes
Problem 13.349 · easy
Find the slope of the line through \( \displaystyle (1, -1) \) and \( \displaystyle (4, -2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-2 + 1}{-1 + 4} = - \frac{1}{3} \]Rise over run.✓ Proved
- \[ \frac{1 - x}{3} - 1 = - \frac{x}{3} - \frac{2}{3} \]Point-slope form through (1, -1), then simplify.✓ Proved
- \[ \left. \frac{\left(-1\right) x}{3} - \frac{2}{3} \right|_{\substack{ x=4 }} = -2 \]It passes through (4, -2) too.✓ Proved
Answer \( m = - \frac{1}{3}\ (\text{decreasing}),\quad y = - \frac{x}{3} - \frac{2}{3} \)
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 both points; its slope matches |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The slope calculation in line 1 is algebraically incorrect; it computes (-2 + 1) / (-1 + 4) = -1/5, not -1/3. The correct slope is -1/3, derived from (-2 - (-1)) / (4 - 1) = -1/3. The solution presents a false equality as proved.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The slope calculation in line 1 is algebraically incorrect; it computes (-2 + 1) / (-1 + 4) = -1/5, not -1/3. The correct slope is -1/3, derived from (-2 - (-1)) / (4 - 1) = -1/3. The solution presents a false equality as proved.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The calculation of the slope in step 1 is incorrect; the formula used is (y1 - y2) / (x2 - x1) or similar mix-up, resulting in -1/3 instead of the cgpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The calculation of the slope is incorrect; the numerator should be the difference in y-values (-2 - (-1) = -1), but the denominator is calculated asgpt-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.