Lines and slopes
Problem 13.424 · easy
Find the slope of the line through \( \displaystyle (3, -4) \) and \( \displaystyle (-1, 5) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{4 + 5}{-3 - 1} = - \frac{9}{4} \]Rise over run.✓ Proved
- \[ - \frac{9 \left(x - 3\right)}{4} - 4 = \frac{11}{4} - \frac{9 x}{4} \]Point-slope form through (3, -4), then simplify.✓ Proved
- \[ \left. - \frac{9 x}{4} + \frac{11}{4} \right|_{\substack{ x=-1 }} = 5 \]It passes through (-1, 5) too.✓ Proved
Answer \( m = - \frac{9}{4}\ (\text{decreasing}),\quad y = \frac{11}{4} - \frac{9 x}{4} \)
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 function as decreasing, and derives the correct slope-intercept equation. The verification step confirms the line passes through the second point.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly calculates the slope, identifies the function as decreasing, and derives the correct slope-intercept equation. The verification step confirms the line passes through the second point.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the line passes through the second point.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: fail (error) 2026-10-21 — The slope calculation in line 1 is incorrect; it computes (4+5)/(-3-1) instead of the correct rise/run (5 - (-4))/(-1 - 3) = 9/-4. Although the final numerical answer is correct, the derivation shown is mathematically invalid.gpt-oss:20b: pass 2026-10-21
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 2026-10-22 with SymPy 1.14.0.