Home›Calculus 1›Lines and slopes›Problem 13.506

Lines and slopes

Problem 13.506 · easy

Find the equation of the line through \( \displaystyle (-2, 5) \) that is parallel to \( \displaystyle y = 4 x - 3 \), in slope-intercept form.

Parallel lines share a slope. The slope is 4.
Reviewed
\[ 4 \left(x + 2\right) + 5 = 4 x + 13 \]
Point-slope form, then simplify.✓ Proved
\[ \left. 4 x + 13 \right|_{\substack{ x=-2 }} = 5 \]
The line passes through (-2, 5).✓ Proved

Answer \( y = 4 x + 13 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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	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 from the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the point lies on the line.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly identifies the slope from the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the point lies on the line.
gpt-oss:20b: pass 2026-11-13
qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly identifies the slope and derives the correct equation. The algebraic checks provided in lines 2 and 3 are valid verifications of the result.
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.