Lines and slopes
Problem 13.136 · easy
Find the equation of the line with slope \( \displaystyle 3 \) through \( \displaystyle (5, -4) \), in slope-intercept form.
- \[ 3 \left(x - 5\right) - 4 = 3 x - 19 \]Point-slope form, then simplify.✓ Proved
- \[ \left. 3 x - 19 \right|_{\substack{ x=5 }} = -4 \]The line passes through (5, -4).✓ Proved
Answer \( y = 3 x - 19 \)
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 line incorrectly applies point‑slope form: it omits the dependent variable y, writing 3*(x-5)-4 instead of y+4 = 3(x-5). This misstates the equation of the line, though the final numeric check happens to be correct.qwen3.6:27b-mlx: fail (misleading) — The solution labels the first equation as 'Point-slope form', but the expression `3*(x - 5) - 4` is not the point-slope form (which is `y - y1 = m(x - x1)`). It is an incorrect intermediate step that happens to simplify to the correct answer, but describing it as the standard form is factually wrong and confusing.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution labels the first equation as 'Point-slope form', but the expression `3*(x - 5) - 4` is not the point-slope form (which is `y - y1 = m(x - x1)`). It is an incorrect intermediate step that happens to simplify to the correct answer, but describing it as the standard form is factually wrong and confusing.gpt-oss:20b: fail (error) 2026-10-17 — The first line incorrectly applies point‑slope form: it omits the dependent variable y, writing 3*(x-5)-4 instead of y+4 = 3(x-5). This misstates the equation of the line, though the final numeric check happens to be correct.qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The solution claims to use point-slope form but presents an incorrect equation `3*(x - 5) - 4 = 3*x - 19` (which simplifies to `-19 = -19`, a tautology, not the line equation). The correct point-slope form is `y - (-4) = 3(x - 5)`. This teaches the student the wrong formula for point-slope form.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly applies point‑slope form: it omits the variable y and equates 3(x‑5)‑4 directly to 3x‑19, which is not a valid equation of the line. The correct form is y+4=3(x‑5) leading to y=3x‑19.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly derives the slope-intercept form and verifies the point. Although the first line is labeled 'unchecked', the algebraic simplification is correct and the verification step confirms the result.gpt-oss:20b: fail (error) 2026-10-14 — The first sentence claims to use point‑slope form, but the equation written is not of the form y−y₁=m(x−x₁). It omits the y‑term entirely and simply equates 3(x−5)−4 to 3x−19, which is not a correct representation of the line. The setup is therefore 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.