Lines and slopes
Problem 13.307 · easy
Find the equation of the line with slope \( \displaystyle 4 \) through \( \displaystyle (-4, -5) \), in slope-intercept form.
- \[ 4 \left(x + 4\right) - 5 = 4 x + 11 \]Point-slope form, then simplify.✓ Proved
- \[ \left. 4 x + 11 \right|_{\substack{ x=-4 }} = -5 \]The line passes through (-4, -5).✓ Proved
Answer \( y = 4 x + 11 \)
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. 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 | ✓ 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: fail (error) — The solution omits the y‑variable in the point‑slope step, effectively writing a tautology 4x+11=4x+11 instead of the correct form y+5=4(x+4). The subsequent substitution checks the y‑value but the derivation is invalid.qwen3.6:27b-mlx: pass — The solution correctly derives the equation using the point-slope form and verifies the result. The algebraic steps are valid and the final answer is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly derives the equation using the point-slope form and verifies the result. The algebraic steps are valid and the final answer is correct.gpt-oss:20b: fail (error) 2026-10-18 — The solution omits the y‑variable in the point‑slope step, effectively writing a tautology 4x+11=4x+11 instead of the correct form y+5=4(x+4). The subsequent substitution checks the y‑value but the derivation is invalid.qwen3.6:27b-mlx: pass 2026-10-16 — 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: fail (error) 2026-10-16 — The solution omits the dependent variable y in the point‑slope equation. It writes 4*(x+4)-5 = 4x+11, which is an identity rather than the correct form y+5 = 4(x+4). The setup is incorrect, so the conclusion is not justified.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution claims to use point-slope form but writes an incorrect equation `4*(x + 4) - 5 = 4*x + 11` which does not represent the line derivation correctly (it equates the LHS of a shifted function to the RHS). The standard point-slope form is `y - y1 = m(x - x1)`, which would be `y + 5 = 4(x + 4)`. The provided equation is algebraically nonsensical as a derivation step.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.