Lines and slopes
Problem 13.122 · easy
Find the slope of the line through \( \displaystyle (3, -4) \) and \( \displaystyle (-3, 3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{3 + 4}{-3 - 3} = - \frac{7}{6} \]Rise over run.✓ Proved
- \[ - \frac{7 \left(x - 3\right)}{6} - 4 = - \frac{7 x}{6} - \frac{1}{2} \]Point-slope form through (3, -4), then simplify.✓ Proved
- \[ \left. - \frac{7 x}{6} - 1 \cdot \frac{1}{2} \right|_{\substack{ x=-3 }} = 3 \]It passes through (-3, 3) too.✓ Proved
Answer \( m = - \frac{7}{6}\ (\text{decreasing}),\quad y = - \frac{7 x}{6} - \frac{1}{2} \)
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 both points; its slope matches |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The algebraic steps are verified and the logic is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The slope calculation is incorrect; the rise is 3 - (-4) = 7, not 3 + 4, although the numerical result -7/6 happens to be correct by coincidence. The explanation 'Rise over run' implies the formula (y2-y1)/(x2-x1), but the numerator written is (3+4) which corresponds to y2 + y1, not y2 - y1.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the second point lies on the line.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.