Lines and slopes
Problem 13.343 · easy
Find the equation of the line with slope \( \displaystyle 2 \) through \( \displaystyle (3, -3) \), in slope-intercept form.
- \[ 2 \left(x - 3\right) - 3 = 2 x - 9 \]Point-slope form, then simplify.✓ Proved
- \[ \left. 2 x - 9 \right|_{\substack{ x=3 }} = -3 \]The line passes through (3, -3).✓ Proved
Answer \( y = 2 x - 9 \)
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) — Step 1 incorrectly applies the point‑slope formula: it omits the variable y and equates 2(x‑3)‑3 directly to 2x‑9, which is not the correct representation of the line. The correct form should be y+3=2(x‑3) leading to y=2x‑9.qwen3.6:27b-mlx: fail (misleading) — The solution lacks the crucial step of stating the point-slope formula and substituting the given values into it. It jumps directly to an algebraic identity without showing the modeling step, which is the core of the problem.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution lacks the crucial step of stating the point-slope formula and substituting the given values into it. It jumps directly to an algebraic identity without showing the modeling step, which is the core of the problem.gpt-oss:20b: fail (error) 2026-10-19 — Step 1 incorrectly applies the point‑slope formula: it omits the variable y and equates 2(x‑3)‑3 directly to 2x‑9, which is not the correct representation of the line. The correct form should be y+3=2(x‑3) leading to y=2x‑9.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly state the equation of the line (y = 2x - 9) as a derived result. It only verifies that the expression 2x - 9 satisfies the point condition, but does not establish that this expression is the y-value of the line. A student would not learn how to derive the equation, only how to check a guess.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the slope-intercept form and verifies the point. The algebraic steps are valid and the final answer is correct.gpt-oss:20b: fail (error) 2026-10-16 — The solution incorrectly applies the point‑slope formula: it omits the variable y on the left side. The correct step should be y+3=2(x-3), which simplifies to y=2x-9. The given line of reasoning is mathematically 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 2026-10-19 with SymPy 1.14.0.