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Lines and slopes

Problem 13.127 · easy

Find the equation of the line through \( \displaystyle (-3, 5) \) that is parallel to \( \displaystyle y = 4 x \), in slope-intercept form.

Parallel lines share a slope. The slope is 4.
Reviewed
\[ 4 \left(x + 3\right) + 5 = 4 x + 17 \]
Point-slope form, then simplify.✓ Proved
\[ \left. 4 x + 17 \right|_{\substack{ x=-3 }} = 5 \]
The line passes through (-3, 5).✓ Proved

Answer \( y = 4 x + 17 \)

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the slope and uses the point-slope form to derive the equation. The algebraic steps are verified, and the final answer is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the slope and uses the point-slope form to derive the equation. The algebraic steps are verified, and the final answer is correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the slope and uses the point-slope form to derive the correct equation. The algebraic steps are verified, and the final answer is correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims to use point-slope form in step 2, but the equation `4*(x + 3) + 5 = 4*x + 17` is an identity used to verify the result, not the derivation itself. The actual point-slope form `y - 5 = 4(x + 3)` is skipped, making the logical flow incomplete and confusing.
gpt-oss:20b: fail (error) 2026-10-14 — The point‑slope step is incorrect: it should be y‑5 = 4(x+3), not 4*(x+3)+5 = 4*x+17. The algebraic manipulation is wrong, leading to an incorrect equation.

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 2026-10-17 with SymPy 1.14.0.