Lines and slopes
Problem 13.342 · easy
Find the slope of the line through \( \displaystyle (-5, -5) \) and \( \displaystyle (-2, -3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-3 + 5}{-2 + 5} = \frac{2}{3} \]Rise over run.✓ Proved
- \[ \frac{2 \left(x + 5\right)}{3} - 5 = \frac{2 x}{3} - \frac{5}{3} \]Point-slope form through (-5, -5), then simplify.✓ Proved
- \[ \left. \frac{2 x}{3} - \frac{5}{3} \right|_{\substack{ x=-2 }} = -3 \]It passes through (-2, -3) too.✓ Proved
Answer \( m = \frac{2}{3}\ (\text{increasing}),\quad y = \frac{2 x}{3} - \frac{5}{3} \)
✓ 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 increasing, 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 2026-10-19 — The solution correctly calculates the slope, identifies the line as increasing, and derives the correct slope-intercept equation. The verification step confirms the line passes through the second point.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly calculates the slope, identifies the line as increasing, and derives the correct slope-intercept equation. The verification step confirms the result.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly calculates the slope, identifies the function as increasing, and derives the correct slope-intercept equation. The verification step confirms the line passes through the second point.gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.