Lines and slopes
Problem 13.130 · easy
Find the equation of the line with slope \( \displaystyle - \frac{3}{2} \) through \( \displaystyle (3, 1) \), in slope-intercept form.
- \[ 1 - \frac{3 \left(x - 3\right)}{2} = \frac{11}{2} - \frac{3 x}{2} \]Point-slope form, then simplify.✓ Proved
- \[ \left. - \frac{3 x}{2} + \frac{11}{2} \right|_{\substack{ x=3 }} = 1 \]The line passes through (3, 1).✓ Proved
Answer \( y = \frac{11}{2} - \frac{3 x}{2} \)
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 omits the variable y in the point‑slope form; it should read y - 1 = -3/2(x - 3) rather than 1 - 3*(x - 3)/2 = …qwen3.6:27b-mlx: fail (misleading) — The solution fails to explicitly state the point-slope formula or the substitution steps, jumping directly to the result. It also presents the final answer in a non-standard form (constant first) without explaining that this is equivalent to the requested slope-intercept form, which could confuse a student expecting y = mx + b.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution fails to explicitly state the point-slope formula or the substitution steps, jumping directly to the result. It also presents the final answer in a non-standard form (constant first) without explaining that this is equivalent to the requested slope-intercept form, which could confuse a student expecting y = mx + b.gpt-oss:20b: fail (error) 2026-10-17 — The first sentence incorrectly omits the variable y in the point‑slope form; it should read y - 1 = -3/2(x - 3) rather than 1 - 3*(x - 3)/2 = …qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution fails to explicitly state the point-slope formula or the substitution step, jumping directly to an algebraic identity. A student would not learn how to derive the equation from the given slope and point.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution provides no derivation or explanation of how the equation was obtained, merely stating the final form and verifying a point. It fails to demonstrate the use of the point-slope formula or the algebraic steps to reach slope-intercept form, which is the core of the problem.gpt-oss:20b: fail (error) 2026-10-14 — The first line incorrectly sets up the point‑slope equation: it should be y-1 = (-3/2)(x-3), not 1-3*(x-3)/2 = 11/2-3*x/2. Consequently the derived line equation is incorrect.
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.