Lines and slopes
Problem 13.346 · easy
Find the equation of the line through \( \displaystyle (2, -2) \) that is parallel to \( \displaystyle y = x - 3 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 1.
- \[ 1 \left(x - 2\right) - 2 = x - 4 \]Point-slope form, then simplify.✓ Proved
- \[ \left. x - 4 \right|_{\substack{ x=2 }} = -2 \]The line passes through (2, -2).✓ Proved
Answer \( y = x - 4 \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope step is incorrectly written: it omits the \(y\) variable and misstates the equation. The correct form is \(y+2=1(x-2)\), which simplifiesqwen3.6:27b-mlx: pass — 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.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — 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: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The point‑slope step is incorrectly written: it omits the \(y\) variable and misstates the equation. The correct form is \(y+2=1(x-2)\), which simplifiesqwen3.6:27b-mlx: pass 2026-10-16 — 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) 2026-10-16 — The second sentence incorrectly applies the point‑slope formula: it omits the y‑term and equates the left side to the right side without showing the derivation of y = x – 4. The algebraic step is not justified and would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly state the point-slope form equation $y - y_1 = m(x - x_1)$ before simplifying, making the transition to the final equation opaque. Additionally, the verification step is presented as an algebraic identity rather than a check of the point, which is confusing.gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the point‑slope formula: it omits the variable \(y\) and incorrectly sets \(1(x-2)-2=x-4\). The subsequent substitution step is al
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 2026-10-19 with SymPy 1.14.0.