Lines and slopes
Problem 13.344 · easy
Find the equation of the line through \( \displaystyle (5, -5) \) that is perpendicular to \( \displaystyle y = 1 - \frac{4 x}{3} \), in slope-intercept form.
- \[ - \frac{1}{\left(-1\right) 4 \cdot \frac{1}{3}} = \frac{3}{4} \]Perpendicular slopes are negative reciprocals.✓ Proved
- \[ \frac{3 \left(x - 5\right)}{4} - 5 = \frac{3 x}{4} - \frac{35}{4} \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{3 x}{4} - \frac{35}{4} \right|_{\substack{ x=5 }} = -5 \]The line passes through (5, -5).✓ Proved
Answer \( y = \frac{3 x}{4} - \frac{35}{4} \)
✓ 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 the point with the required slope |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the negative reciprocal slope and uses the point-slope form to derive the correct equation. The verification step confirms the line passes through the given point.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the negative reciprocal slope and uses the point-slope form to derive the correct equation. The verification step confirms the line passes through the given point.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the negative reciprocal slope and applies the point-slope formula accurately. The verification step confirms the line passes through the given point.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the negative reciprocal slope and applies the point-slope formula accurately. The verification step confirms the line passes through the given 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.