Lines and slopes
Problem 13.507 · easy
Find the equation of the line with slope \( \displaystyle \frac{2}{3} \) through \( \displaystyle (-4, 4) \), in slope-intercept form.
- \[ \frac{2 \left(x + 4\right)}{3} + 4 = \frac{2 x}{3} + \frac{20}{3} \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{2 x}{3} + \frac{20}{3} \right|_{\substack{ x=-4 }} = 4 \]The line passes through (-4, 4).✓ Proved
Answer \( y = \frac{2 x}{3} + \frac{20}{3} \)
✓ 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the line meets the point with the required slope |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly derives the slope-intercept form and verifies the point. The steps are logically sound and algebraically correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly derives the slope-intercept form and verifies the point. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-11-13qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly derives the slope-intercept form and verifies the point. Although labeled 'unchecked', the algebraic steps are correct and consistent with the problem statement.gpt-oss:20b: pass 2026-11-13
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 2026-11-13 with SymPy 1.14.0.