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Lines and slopes

Problem 13.421 · easy

Find the slope of the line through \( \displaystyle (1, 3) \) and \( \displaystyle (-1, 1) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.

\[ \frac{-3 + 1}{-1 - 1} = 1 \]
Rise over run.✓ Proved
\[ 1 \left(x - 1\right) + 3 = x + 2 \]
Point-slope form through (1, 3), then simplify.✓ Proved
\[ \left. x + 2 \right|_{\substack{ x=-1 }} = 1 \]
It passes through (-1, 1) too.✓ Proved

Answer \( m = 1\ (\text{increasing}),\quad y = x + 2 \)

✓ 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 both points; its slope matches

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly calculates the slope, identifies the function as increasing, and derives the correct equation in slope-intercept form. The algebraic steps are verified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly calculates the slope, identifies the function as increasing, and derives the correct equation in slope-intercept form. The algebraic steps are verified.
gpt-oss:20b: pass 2026-10-22
qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly calculates the slope, identifies the function as increasing, and derives the correct slope-intercept equation. The steps are logically sound and algebraically verified.
gpt-oss:20b: pass 2026-10-21
qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly calculates the slope, identifies the function as increasing, and derives the correct slope-intercept equation. The verification step confirms the line passes through the second point.
gpt-oss:20b: pass 2026-10-21

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 2026-10-22 with SymPy 1.14.0.