Lines and slopes
Problem 13.425 · easy
Find the equation of the line through \( \displaystyle (0, -1) \) that is parallel to \( \displaystyle y = \frac{3 x}{2} - 2 \), in slope-intercept form.
- Parallel lines share a slope. The slope is 3/2.
- \[ \frac{3 \left(x + 0\right)}{2} - 1 = \frac{3 x}{2} - 1 \]Point-slope form, then simplify.✓ Proved
- \[ \left. \frac{3 x}{2} - 1 \right|_{\substack{ x=0 }} = -1 \]The line passes through (0, -1).✓ Proved
Answer \( y = \frac{3 x}{2} - 1 \)
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: fail (error) — The solution incorrectly applies point‑slope form: it omits the y‑term and uses 3*(x+0)/2 – 1 instead of y+1 = (3/2)(x–0).qwen3.6:27b-mlx: pass — The solution correctly identifies the slope from the parallel line and uses the given point to determine the y-intercept. The algebraic steps are valid and lead to the correct slope-intercept form.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly identifies the slope from the parallel line and uses the given point to determine the y-intercept. The algebraic steps are valid and lead to the correct slope-intercept form.gpt-oss:20b: fail (error) 2026-10-22 — The solution incorrectly applies point‑slope form: it omits the y‑term and uses 3*(x+0)/2 – 1 instead of y+1 = (3/2)(x–0).qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly identifies the slope from the parallel line and uses the given point to determine the y-intercept. The algebraic steps are valid and lead to the correct slope-intercept form.gpt-oss:20b: fail (error) 2026-10-21 — The solution incorrectly applies the point‑slope formula. It omits the left‑hand side y‑y₁ and therefore does not correctly express the line. The resulting equation is not justified by the steps shown.qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly identifies the slope from the parallel line and uses the given point to determine the y-intercept. The final equation is correct.gpt-oss:20b: fail (error) 2026-10-21 — The solution incorrectly applies point‑slope form. It omits the left‑hand side (y+1) and therefore does not correctly derive the line equation.
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 2026-10-22 with SymPy 1.14.0.