Home›Calculus 1›Lines and slopes›Problem 13.501

Lines and slopes

Problem 13.501 · easy

Find the equation of the line with slope \( \displaystyle - \frac{1}{2} \) through \( \displaystyle (4, -2) \), in slope-intercept form.

\[ \frac{4 - x}{2} - 2 = - \frac{x}{2} \]
Point-slope form, then simplify.✓ Proved
\[ \left. \frac{\left(-1\right) x}{2} \right|_{\substack{ x=4 }} = -2 \]
The line passes through (4, -2).✓ Proved

Answer \( y = - \frac{x}{2} \)

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. A reviewer's objection to the explanation stands; see 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The first line misapplies point‑slope form: it should be \(y+2=-\frac12(x-4)\), which simplifies to \(y=-\frac{x}{2}\). The given equation is incorrect an
qwen3.6:27b-mlx: fail (error) — The stated answer y = -x/2 does not pass through the point (4, -2), as substituting x=4 yields y=-2, which is correct, but the slope is -1/2. Wait, let me re-evaluate. If y = -x/2, then at x=4, y=-2. The slope is -1/2. So the answer is actually correct. Let me re-read the solution steps. Step 1 shows the derivation from point-slope form: y - (-2) = -1/2(x - 4) => y + 2 = -x/2 + 2 => y = -x/2. This is correct. Step 2 verifies the point. The solution is correct. Why did I think it was wrong? Ah, I misread the verification in my head. Let's look closer. Is there a subtle error? The problem asks for slope-intercept form. y = -x/2 is in slope-intercept form (y = mx + b with b=0). The derivation is correct. The verification is correct. The answer is correct. So it should pass. Let me double check the 'adversarial' nature. Is there anything misleading? No. It's a standard correct solution.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-11-13 — The stated answer y = -x/2 does not pass through the point (4, -2), as substituting x=4 yields y=-2, which is correct, but the slope is -1/2. Wait, let me re-evaluate. If y = -x/2, then at x=4, y=-2. The slope is -1/2. So the answer is actually correct. Let me re-read the solution steps. Step 1 shows the derivation from point-slope form: y - (-2) = -1/2(x - 4) => y + 2 = -x/2 + 2 => y = -x/2. This is correct. Step 2 verifies the point. The solution is correct. Why did I think it was wrong? Ah, I misread the verification in my head. Let's look closer. Is there a subtle error? The problem asks for slope-intercept form. y = -x/2 is in slope-intercept form (y = mx + b with b=0). The derivation is correct. The verification is correct. The answer is correct. So it should pass. Let me double check the 'adversarial' nature. Is there anything misleading? No. It's a standard correct solution.
gpt-oss:20b: inconclusive 2026-11-13 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The first line misapplies point‑slope form: it should be \(y+2=-\frac12(x-4)\), which simplifies to \(y=-\frac{x}{2}\). The given equation is incorrect an
qwen3.6:27b-mlx: fail (error) 2026-11-13 — The solution fails to derive the equation of the line, instead merely verifying that the point (4, -2) satisfies the final answer. It does not show the calculation of the y-intercept or the algebraic steps to reach slope-intercept form.
gpt-oss:20b: fail (misleading) 2026-11-13 — The first sentence presents the point‑slope expression as "-(x-4)/2 - 2 = -x/2" without explicitly introducing the variable y, making the step unclear and potentially confusing a student about the correct form of the 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 2026-11-13 with SymPy 1.14.0.