Lines and slopes
Problem 13.504 · easy
Find the slope of the line through \( \displaystyle (-1, 5) \) and \( \displaystyle (-5, -3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-5 - 3}{-5 + 1} = 2 \]Rise over run.✓ Proved
- \[ 2 \left(x + 1\right) + 5 = 2 x + 7 \]Point-slope form through (-1, 5), then simplify.✓ Proved
- \[ \left. 2 x + 7 \right|_{\substack{ x=-5 }} = -3 \]It passes through (-5, -3) too.✓ Proved
Answer \( m = 2\ (\text{increasing}),\quad y = 2 x + 7 \)
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 calculation in step 1 is incorrect; it computes (-8)/(-4) = 2, but the correct slope is (-3 - 5)/(-5 - (-1)) = -8/-4 = 2. Wait, the calculation is actually correct numerically, but the text 'Rise over run' is misleading because the numerator is written as (-5 - 3) which mixes x and y coordinates incorrectly (it should be y2-y1 or y1-y2). Let's re-read carefully. Point 1: (-1, 5), Point 2: (-5, -3). Slope m = (y2 - y1) / (x2 - x1) = (-3 - 5) / (-5 - (-1)) = -8 / -4 = 2. The solution writes (-5 - 3)/(-5 + 1). The numerator -5 - 3 = -8. The denominator -5 + 1 = -4. The numbers happen to match the correct difference if one interprets -5 as y2? No, y2 is -3. If one interprets -5 as x2, then x2-x1 = -5 - (-1) = -4. So the denominator is correct for x2-x1. The numerator is -5 - 3. This looks like x2 - y1? Or maybe it's just a typo for -3 - 5? -3 - 5 = -8. -5 - 3 = -8. It's a coincidence that the numbers work out. However, the expression (-5 - 3) does not represent the change in y. The change in y is -3 - 5 = -8. The expression used is -5 - 3. This is technically 'error' in the setup of the formula, even if the arithmetic result is correct by coincidence. A student would learn that you can just pick numbers from the points arbitrarily as long as they sum to the right thing? No, they would be confused. It is an error in the application of the slope formula.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-11-13 — The calculation in step 1 is incorrect; it computes (-8)/(-4) = 2, but the correct slope is (-3 - 5)/(-5 - (-1)) = -8/-4 = 2. Wait, the calculation is actually correct numerically, but the text 'Rise over run' is misleading because the numerator is written as (-5 - 3) which mixes x and y coordinates incorrectly (it should be y2-y1 or y1-y2). Let's re-read carefully. Point 1: (-1, 5), Point 2: (-5, -3). Slope m = (y2 - y1) / (x2 - x1) = (-3 - 5) / (-5 - (-1)) = -8 / -4 = 2. The solution writes (-5 - 3)/(-5 + 1). The numerator -5 - 3 = -8. The denominator -5 + 1 = -4. The numbers happen to match the correct difference if one interprets -5 as y2? No, y2 is -3. If one interprets -5 as x2, then x2-x1 = -5 - (-1) = -4. So the denominator is correct for x2-x1. The numerator is -5 - 3. This looks like x2 - y1? Or maybe it's just a typo for -3 - 5? -3 - 5 = -8. -5 - 3 = -8. It's a coincidence that the numbers work out. However, the expression (-5 - 3) does not represent the change in y. The change in y is -3 - 5 = -8. The expression used is -5 - 3. This is technically 'error' in the setup of the formula, even if the arithmetic result is correct by coincidence. A student would learn that you can just pick numbers from the points arbitrarily as long as they sum to the right thing? No, they would be confused. It is an error in the application of the slope formula.gpt-oss:20b: pass 2026-11-13qwen3.6:27b-mlx: fail (error) 2026-11-13 — The slope calculation in step 1 is arithmetically incorrect: (-5 - 3)/(-5 + 1) evaluates to -2/(-4) = 0.5, not 2. The correct slope is 2, derived from (-3 - 5)/(-5 - (-1)) = -8/-4 = 2. The solution presents a false equality as a proven fact.gpt-oss:20b: pass 2026-11-13
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.