Lines and slopes
Problem 13.300 · easy
Find the slope of the line through \( \displaystyle (5, 5) \) and \( \displaystyle (-4, 0) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-5 + 0}{-5 - 4} = \frac{5}{9} \]Rise over run.✓ Proved
- \[ \frac{5 \left(x - 5\right)}{9} + 5 = \frac{5 x}{9} + \frac{20}{9} \]Point-slope form through (5, 5), then simplify.✓ Proved
- \[ \left. \frac{5 x}{9} + \frac{20}{9} \right|_{\substack{ x=-4 }} = 0 \]It passes through (-4, 0) too.✓ Proved
Answer \( m = \frac{5}{9}\ (\text{increasing}),\quad y = \frac{5 x}{9} + \frac{20}{9} \)
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 step 1 uses incorrect coordinates (-5 instead of 5 for the y-values), resulting in the wrong sign for the slope. The correct slope is -5/9, making the line decreasing, not increasing.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The slope calculation in step 1 uses incorrect coordinates (-5 instead of 5 for the y-values), resulting in the wrong sign for the slope. The correct slope is -5/9, making the line decreasing, not increasing.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The slope calculation in step 1 uses coordinates (-5, 0) and (-5, 4) or similar incorrect values, rather than the given points (5, 5) and (-4, 0). The correct slope is 5/9, but the derivation shown is algebraically inconsistent with the problem statement's points (e.g., using -5 instead of 5 for the x-coordinate of the first point). While the final numerical answer is correct, the work shown is wrong.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The slope calculation in step 1 uses incorrect coordinates (-5 instead of 5), resulting in the wrong slope m = 5/9 instead of the correct m = 5/9? Wgpt-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.