Lines and slopes
Problem 13.500 · easy
Find the slope of the line through \( \displaystyle (5, 0) \) and \( \displaystyle (1, -4) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-4 + 0}{-5 + 1} = 1 \]Rise over run.✓ Proved
- \[ 1 \left(x - 5\right) + 0 = x - 5 \]Point-slope form through (5, 0), then simplify.✓ Proved
- \[ \left. x - 5 \right|_{\substack{ x=1 }} = -4 \]It passes through (1, -4) too.✓ Proved
Answer \( m = 1\ (\text{increasing}),\quad y = x - 5 \)
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 |
| 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 both points; its slope matches |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The slope calculation in step 1 is incorrect; the numerator should be -4 - 0 = -4 and the denominator 1 - 5 = -4, yielding a slope of 1, but the written expression (-4 + 0)/(-5 + 1) evaluates to -4/-4 = 1, which is numerically correct but the setup implies adding the y-coordinates and subtracting x-coordinates incorrectly (standard formula is (y2-y1)/(x2-x1)). Wait, let's re-evaluate. (-4 - 0) / (1 - 5) = -4 / -4 = 1. The solution writes (-4 + 0)/(-5 + 1). This is (-4)/(-4) = 1. This is actually correct arithmetic for the points (5,0) and (1,-4) if interpreted as (y2 - y1)/(x2 - x1) where y2=-4, y1=0, x2=1, x1=5? No, x2-x1 = 1-5 = -4. The solution writes -5+1 which is -4. So the denominator is correct. The numerator is -4+0 which is -4. So the calculation is correct. However, the text says 'Rise over run'. Usually rise is y2-y1. Here it is -4 - 0. The solution writes -4 + 0. This is equivalent to -(4-0) or -4. It's a bit weird notation but mathematically valid. Let's look closer. Is there a bigger error? The problem asks for slope, increasing/decreasing, and equation. The solution provides m=1, increasing, y=x-5. All correct. Why would it fail? Let's check the sentence 'Rise over run'. It's a label. Is it misleading? No. Let's check the algebra again. (-4+0)/(-5+1) = -4/-4 = 1. Correct. Point slope: y - 0 = 1(x - 5) => y = x - 5. Correct. Check point (1, -4): -4 = 1 - 5. Correct. The solution seems correct. Wait, look at step 1 again. `(-4 + 0)/(-5 + 1)`. If the points are (5,0) and (1,-4), the standard formula is (y2-y1)/(x2-x1). If we take (x1,y1)=(5,0) and (x2,y2)=(1,-4), then y2-y1 = -4-0 = -4. x2-x1 = 1-5 = -4. The solution writes `-4 + 0` for the numerator. This is `-4`. It writes `-5 + 1` for the denominator. This is `-4`. So the numbers are right. But is the *setup* described by the sentence 'Rise over run' accurate? Yes. Is there any other issue? Maybe the order of points? If we did (0 - (-4)) / (5 - 1) = 4/4 = 1. The solution used the other order. Both are valid. The result is correct. The steps are correct. The conclusion is correct. I will mark it as pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-11-13 — The slope calculation in step 1 is incorrect; the numerator should be -4 - 0 = -4 and the denominator 1 - 5 = -4, yielding a slope of 1, but the written expression (-4 + 0)/(-5 + 1) evaluates to -4/-4 = 1, which is numerically correct but the setup implies adding the y-coordinates and subtracting x-coordinates incorrectly (standard formula is (y2-y1)/(x2-x1)). Wait, let's re-evaluate. (-4 - 0) / (1 - 5) = -4 / -4 = 1. The solution writes (-4 + 0)/(-5 + 1). This is (-4)/(-4) = 1. This is actually correct arithmetic for the points (5,0) and (1,-4) if interpreted as (y2 - y1)/(x2 - x1) where y2=-4, y1=0, x2=1, x1=5? No, x2-x1 = 1-5 = -4. The solution writes -5+1 which is -4. So the denominator is correct. The numerator is -4+0 which is -4. So the calculation is correct. However, the text says 'Rise over run'. Usually rise is y2-y1. Here it is -4 - 0. The solution writes -4 + 0. This is equivalent to -(4-0) or -4. It's a bit weird notation but mathematically valid. Let's look closer. Is there a bigger error? The problem asks for slope, increasing/decreasing, and equation. The solution provides m=1, increasing, y=x-5. All correct. Why would it fail? Let's check the sentence 'Rise over run'. It's a label. Is it misleading? No. Let's check the algebra again. (-4+0)/(-5+1) = -4/-4 = 1. Correct. Point slope: y - 0 = 1(x - 5) => y = x - 5. Correct. Check point (1, -4): -4 = 1 - 5. Correct. The solution seems correct. Wait, look at step 1 again. `(-4 + 0)/(-5 + 1)`. If the points are (5,0) and (1,-4), the standard formula is (y2-y1)/(x2-x1). If we take (x1,y1)=(5,0) and (x2,y2)=(1,-4), then y2-y1 = -4-0 = -4. x2-x1 = 1-5 = -4. The solution writes `-4 + 0` for the numerator. This is `-4`. It writes `-5 + 1` for the denominator. This is `-4`. So the numbers are right. But is the *setup* described by the sentence 'Rise over run' accurate? Yes. Is there any other issue? Maybe the order of points? If we did (0 - (-4)) / (5 - 1) = 4/4 = 1. The solution used the other order. Both are valid. The result is correct. The steps are correct. The conclusion is correct. I will mark it as pass.gpt-oss:20b: pass 2026-11-13qwen3.6:27b-mlx: inconclusive 2026-11-13 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The slope calculation in step 1 is incorrect; it computes (-4 - 0) / (1 - 5) = 4/4 = 1, but the text shows (-4 + 0)/(-5 + 1) which implies a denomingpt-oss:20b: fail (error) 2026-11-13 — Step 2 incorrectly omits the dependent variable y; the point‑slope form should be y-0=1(x-5), leading to y=x-5. The solution as written does not correctly express 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 2026-11-13 with SymPy 1.14.0.