Lines and slopes
Problem 13.141 · easy
Find the equation of the line through \( \displaystyle (3, -4) \) that is parallel to \( \displaystyle y = x - 4 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 1.
- \[ 1 \left(x - 3\right) - 4 = x - 7 \]Point-slope form, then simplify.✓ Proved
- \[ \left. x - 7 \right|_{\substack{ x=3 }} = -4 \]The line passes through (3, -4).✓ Proved
Answer \( y = x - 7 \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 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 the point with the required slope |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — Line 2 presents a nonsensical algebraic identity (1*(x - 3) - 4 = x - 7) as the derivation step, rather than showing the standard point-slope form y - y1 = m(x - x1). This obscures the actual method and teaches poor notation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — Line 2 presents a nonsensical algebraic identity (1*(x - 3) - 4 = x - 7) as the derivation step, rather than showing the standard point-slope form y - y1 = m(x - x1). This obscures the actual method and teaches poor notation.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the slope of the parallel line and uses the point-slope form to derive the correct equation. The verification step confirms the point lies on the line.gpt-oss:20b: fail (error) 2026-10-15 — Sentence 2 incorrectly applies point‑slope form; it omits the y‑variable and equates an expression to itself instead of deriving y = x – 7.qwen3.6:27b-mlx: fail (error) 2026-10-14 — Line 2 presents an algebraically invalid equation '1*(x - 3) - 4 = x - 7' as a step in the derivation. The correct point-slope form is y - (-4) = 1*(x - 3), which simplifies to y = x - 7. The provided equation implies -4 = -7 if x is substituted, or is simply a malformed equality that does not represent the line equation correctly.gpt-oss:20b: fail (error) 2026-10-14 — The point‑slope step is incorrect: it should be y+4=1(x−3), leading to y=x−7. The given equation 1*(x−3)−4=x−7 is algebraically invalid.
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.