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Lines and slopes

Problem 13.121 · easy

Find the equation of the line with slope \( \displaystyle \frac{5}{2} \) through \( \displaystyle (4, 2) \), in slope-intercept form.

\[ \frac{5 \left(x - 4\right)}{2} + 2 = \frac{5 x}{2} - 8 \]
Point-slope form, then simplify.✓ Proved
\[ \left. \frac{5 x}{2} - 8 \right|_{\substack{ x=4 }} = 2 \]
The line passes through (4, 2).✓ Proved

Answer \( y = \frac{5 x}{2} - 8 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are valid and the final answer matches the problem statement.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are valid and the final answer matches the problem statement.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are valid and the final answer is correct.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are sound.
gpt-oss:20b: pass 2026-10-14

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.