Lines and slopes
Problem 13.298 · easy
Find the slope of the line through \( \displaystyle (3, -5) \) and \( \displaystyle (0, 0) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{0 + 5}{-3 + 0} = - \frac{5}{3} \]Rise over run.✓ Proved
- \[ - \frac{5 \left(x - 3\right)}{3} - 5 = - \frac{5 x}{3} \]Point-slope form through (3, -5), then simplify.✓ Proved
- \[ \left. - \frac{5 x}{3} \right|_{\substack{ x=0 }} = 0 \]It passes through (0, 0) too.✓ Proved
Answer \( m = - \frac{5}{3}\ (\text{decreasing}),\quad y = - \frac{5 x}{3} \)
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: fail (style) — The solution omits the required sentence stating that the line is decreasing (since the slope is negative).qwen3.6:27b-mlx: fail (error) — The calculation of the slope in line 1 is incorrect; it computes (y2-y1)/(x1-x2) instead of (y2-y1)/(x2-x1), resulting in the wrong sign. The correct slope is 5/3.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The calculation of the slope in line 1 is incorrect; it computes (y2-y1)/(x1-x2) instead of (y2-y1)/(x2-x1), resulting in the wrong sign. The correct slope is 5/3.gpt-oss:20b: fail (style) 2026-10-18 — The solution omits the required sentence stating that the line is decreasing (since the slope is negative).qwen3.6:27b-mlx: fail (error) 2026-10-16 — The calculation for the slope in step 1 is incorrect; it computes (0 - (-5)) / (-3 - 0) = -5/3, which corresponds to the points (0,0) and (-3,0), not (3,-5) and (0,0). The correct slope is (-5 - 0) / (3 - 0) = -5/3, but the setup shown is mathematically inconsistent with the given points.gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 2 incorrectly states the point‑slope form. The correct form is \(y+5=\frac{-5}{3}(x-3)\), which simplifies to \(y=-\frac{5}{3}x\). The given equationqwen3.6:27b-mlx: fail (error) 2026-10-15 — The slope calculation in line 1 is algebraically incorrect; it computes (0+5)/(-3+0) = -5/3, but the correct rise over run for points (3, -5) and (0, 0) is (-5-0)/(3-0) = -5/3. While the numerical result is coincidentally correct, the formula used (adding coordinates instead of subtracting) is fundamentally wrong and would fail for most other point pairs. Additionally, the explanation 'Rise over run' does not justify the erroneous arithmetic.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.