Lines and slopes
Problem 13.296 · easy
Find the slope of the line through \( \displaystyle (-5, -1) \) and \( \displaystyle (4, -2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-2 + 1}{4 + 5} = - \frac{1}{9} \]Rise over run.✓ Proved
- \[ \frac{- x - 5}{9} - 1 = - \frac{x}{9} - \frac{14}{9} \]Point-slope form through (-5, -1), then simplify.✓ Proved
- \[ \left. \frac{\left(-1\right) x}{9} - \frac{14}{9} \right|_{\substack{ x=4 }} = -2 \]It passes through (4, -2) too.✓ Proved
Answer \( m = - \frac{1}{9}\ (\text{decreasing}),\quad y = - \frac{x}{9} - \frac{14}{9} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 verification step confirms the result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the result.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly calculates the slope, identifies the function as decreasing, and derives the correct slope-intercept form. The verification step confirms the line passes through the second point.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the result.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.