Home›Calculus 1›Lines and slopes›Problem 13.502

Lines and slopes

Problem 13.502 · easy

Find the slope of the line through \( \displaystyle (-5, 2) \) and \( \displaystyle (-2, -4) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.

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

Answer \( m = -2\ (\text{decreasing}),\quad y = - 2 x - 8 \)

✓ 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
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 line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly calculates the slope, identifies the line as decreasing, and derives the correct slope-intercept equation. The verification step confirms the result.
gpt-oss:20b: pass 2026-11-13
qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly calculates the slope, identifies the function as decreasing, and derives the correct slope-intercept equation. The verification step confirms the line passes through the second point.
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.