Lines and slopes
Problem 13.132 · easy
Find the slope of the line through \( \displaystyle (3, -2) \) and \( \displaystyle (2, 0) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{0 + 2}{-3 + 2} = -2 \]Rise over run.✓ Proved
- \[ - 2 \left(x - 3\right) - 2 = 4 - 2 x \]Point-slope form through (3, -2), then simplify.✓ Proved
- \[ \left. 4 - 2 x \right|_{\substack{ x=2 }} = 0 \]It passes through (2, 0) too.✓ Proved
Answer \( m = -2\ (\text{decreasing}),\quad y = 4 - 2 x \)
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 calculation of the slope in line 1 is incorrect; it computes (0 - (-2)) / (-3 - 2) = -2/5, but the problem requires the slope through (3, -2) and (2, 0), which is (0 - (-2)) / (2 - 3) = -2. The line 1 uses x-coordinates -3 and 2 instead of 3 and 2, implying a typo in the setup or coordinates used, yet claims to be 'Rise over run' for the given points. Although the final numerical answer m=-2 is correct for the stated points, the derivation shown is for different points (-3, 0) and (2, -2) or similar, making the reasoning invalid for the stated problem.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The calculation of the slope in line 1 is incorrect; it computes (0 - (-2)) / (-3 - 2) = -2/5, but the problem requires the slope through (3, -2) and (2, 0), which is (0 - (-2)) / (2 - 3) = -2. The line 1 uses x-coordinates -3 and 2 instead of 3 and 2, implying a typo in the setup or coordinates used, yet claims to be 'Rise over run' for the given points. Although the final numerical answer m=-2 is correct for the stated points, the derivation shown is for different points (-3, 0) and (2, -2) or similar, making the reasoning invalid for the stated problem.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The calculation of the slope in line 1 uses incorrect coordinates (0 and -3 instead of -2 and 2), resulting in the wrong slope value. The correct slope is 2, not -2.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The calculation of the slope in step 1 is incorrect; it computes (y2+y1)/(x2-x1) instead of (y2-y1)/(x2-x1). Although the final numerical answer happens to be correct due to a coincidence in the arithmetic, the method demonstrated is fundamentally wrong.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.