Lines and slopes
Problem 13.302 · easy
Find the equation of the line through \( \displaystyle (3, -1) \) that is parallel to \( \displaystyle y = 5 x \), in slope-intercept form.
- Parallel lines share a slope. The slope is 5.
- \[ 5 \left(x - 3\right) - 1 = 5 x - 16 \]Point-slope form, then simplify.✓ Proved
- \[ \left. 5 x - 16 \right|_{\substack{ x=3 }} = -1 \]The line passes through (3, -1).✓ Proved
Answer \( y = 5 x - 16 \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 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: fail (error) — The solution omits the y‑term in the point‑slope step, so the equation is not derived correctly from the given point and slope. The algebraic manipulation is correct, but the sentence claiming to use point‑slope form is misleading and incomplete.qwen3.6:27b-mlx: pass — The solution correctly identifies the slope and verifies the point. Although the algebraic manipulation in step 2 is presented as a single equality rather than a derivation, the logic is sound and the final answer is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the slope and verifies the point. Although the algebraic manipulation in step 2 is presented as a single equality rather than a derivation, the logic is sound and the final answer is correct.gpt-oss:20b: fail (error) 2026-10-18 — The solution omits the y‑term in the point‑slope step, so the equation is not derived correctly from the given point and slope. The algebraic manipulation is correct, but the sentence claiming to use point‑slope form is misleading and incomplete.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the slope and uses the point-slope form to derive the equation. The algebraic steps are verified, and the final check confirms the line passes through the given point.gpt-oss:20b: fail (error) 2026-10-16 — The point‑slope step omits the variable y; it should read y+1=5(x-3) leading to y=5x-16. The current equation 5*(x-3)-1=5*x-16 is incomplete and misleading.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly state the point-slope equation y - (-1) = 5(x - 3) before simplifying. Instead, it presents an identity check (5*(x-3)-1 = 5x-16) as the derivation step, which is logically circular and does not demonstrate how the equation was derived from the given point and slope.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.