Lines and slopes
Problem 13.503 · easy
Find the equation of the line with slope \( \displaystyle -4 \) through \( \displaystyle (0, 5) \), in slope-intercept form.
- \[ 5 - 4 \left(x + 0\right) = 5 - 4 x \]Point-slope form, then simplify.✓ Proved
- \[ \left. 5 - 4 x \right|_{\substack{ x=0 }} = 5 \]The line passes through (0, 5).✓ Proved
Answer \( y = 5 - 4 x \)
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 first sentence incorrectly applies the point‑slope formula; it produces an identity rather than the line equation y = -4x + 5. The solution does not correctly model the problem.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the point-slope formula or the slope-intercept form derivation, jumping directly to an algebraic simplification without modeling the problem setup. It also lacks a concluding sentence stating the final equation in the requested form.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-11-13 — The solution fails to explicitly state the point-slope formula or the slope-intercept form derivation, jumping directly to an algebraic simplification without modeling the problem setup. It also lacks a concluding sentence stating the final equation in the requested form.gpt-oss:20b: fail (error) 2026-11-13 — The first sentence incorrectly applies the point‑slope formula; it produces an identity rather than the line equation y = -4x + 5. The solution does not correctly model the problem.qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly applies the point-slope formula and verifies the result. The steps are logically sound and lead to the correct slope-intercept form.gpt-oss:20b: fail (error) 2026-11-13 — The solution misapplies the point‑slope formula: it writes 5-4*(x+0)=5-4x, omitting the variable y entirely. The correct point‑slope form is y-5=-4(x-0), which simplifies to y=-4x+5. The subsequent substitution step is therefore based on an incorrect expression.
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.