Lines and slopes
Problem 13.350 · easy
Find the slope of the line through \( \displaystyle (-1, 2) \) and \( \displaystyle (3, 1) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-2 + 1}{1 + 3} = - \frac{1}{4} \]Rise over run.✓ Proved
- \[ 2 - \frac{x + 1}{4} = \frac{7}{4} - \frac{x}{4} \]Point-slope form through (-1, 2), then simplify.✓ Proved
- \[ \left. - \frac{x}{4} + \frac{7}{4} \right|_{\substack{ x=3 }} = 1 \]It passes through (3, 1) too.✓ Proved
Answer \( m = - \frac{1}{4}\ (\text{decreasing}),\quad y = \frac{7}{4} - \frac{x}{4} \)
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; it computes (y2 - y1) / (x2 + x1) instead of (y2 - y1) / (x2 - x1). The correct slope is -1/4, but the derivation shown is mathematically invalid (using 1+3 instead of 1-(-1) or 3-(-1)). While the final numerical answer for the slope happens to be correct due to a coincidence in the numbers (4 vs 4), the method demonstrated is wrong. Specifically, the denominator should be 3 - (-1) = 4, not 1 + 3 = 4. Wait, 1+3 is 4 and 3-(-1) is 4. The numbers are the same, but the formula is wrong. The sentence says 'Rise over run'. Rise is 1-2 = -1. Run is 3-(-1) = 4. The equation shows (-2+1)/(1+3). -2+1 is -1. 1+3 is 4. So the arithmetic is -1/4. But the 'run' is calculated as 1+3. The x-coordinates are -1 and 3. The run is 3 - (-1) = 4. The expression 1+3 suggests the author might have confused the coordinates or used a wrong formula like x2 + x1? No, x1 is -1, x2 is 3. x2 - x1 = 4. x2 + x1 = 2. Wait. 1+3? Where did 1 and 3 come from? The points are (-1, 2) and (3, 1). y1=2, y2=1. x1=-1, x2=3. Rise = 1-2 = -1. Run = 3-(-1) = 4. The equation is (-2+1)/(1+3). Numerator: -2+1 = -1. This looks like y1 is treated as 2, so -y1 + y2? Or y2 - y1? 1-2 = -1. The term is -2+1. That is y2 - y1 if y2=1, y1=2. Denominator: 1+3. This is not x2 - x1 (which is 3 - (-1) = 4). It is not x2 + x1 (which is 3 + (-1) = 2). It is 1+3. Where do 1 and 3 come from? They are the y-coordinate of the second point and the x-coordinate of the second point? Or maybe the absolute values? It's a nonsensical formula for 'run'. It just happens that 1+3=4 and 3-(-1)=4. This is a 'error' because the method is wrong, even if the number is right by coincidence. A student would learn to add coordinates to find run, which is false.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The slope calculation in step 1 is incorrect; it computes (y2 - y1) / (x2 + x1) instead of (y2 - y1) / (x2 - x1). The correct slope is -1/4, but the derivation shown is mathematically invalid (using 1+3 instead of 1-(-1) or 3-(-1)). While the final numerical answer for the slope happens to be correct due to a coincidence in the numbers (4 vs 4), the method demonstrated is wrong. Specifically, the denominator should be 3 - (-1) = 4, not 1 + 3 = 4. Wait, 1+3 is 4 and 3-(-1) is 4. The numbers are the same, but the formula is wrong. The sentence says 'Rise over run'. Rise is 1-2 = -1. Run is 3-(-1) = 4. The equation shows (-2+1)/(1+3). -2+1 is -1. 1+3 is 4. So the arithmetic is -1/4. But the 'run' is calculated as 1+3. The x-coordinates are -1 and 3. The run is 3 - (-1) = 4. The expression 1+3 suggests the author might have confused the coordinates or used a wrong formula like x2 + x1? No, x1 is -1, x2 is 3. x2 - x1 = 4. x2 + x1 = 2. Wait. 1+3? Where did 1 and 3 come from? The points are (-1, 2) and (3, 1). y1=2, y2=1. x1=-1, x2=3. Rise = 1-2 = -1. Run = 3-(-1) = 4. The equation is (-2+1)/(1+3). Numerator: -2+1 = -1. This looks like y1 is treated as 2, so -y1 + y2? Or y2 - y1? 1-2 = -1. The term is -2+1. That is y2 - y1 if y2=1, y1=2. Denominator: 1+3. This is not x2 - x1 (which is 3 - (-1) = 4). It is not x2 + x1 (which is 3 + (-1) = 2). It is 1+3. Where do 1 and 3 come from? They are the y-coordinate of the second point and the x-coordinate of the second point? Or maybe the absolute values? It's a nonsensical formula for 'run'. It just happens that 1+3=4 and 3-(-1)=4. This is a 'error' because the method is wrong, even if the number is right by coincidence. A student would learn to add coordinates to find run, which is false.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The slope calculation in step 1 is incorrect; it computes (y1 - y2) / (x2 + x1) instead of (y2 - y1) / (x2 - x1). The correct slope is -1/4, but the arithmetic shown (-2 + 1)/(1 + 3) equals -1/4 only by coincidence of signs if interpreted as (y1-y2)/(x1+x2) which is not the formula, or if the points were swapped incorrectly. Actually, (-2+1)/(1+3) = -1/4. Let's check the real slope: (1-2)/(3-(-1)) = -1/4. The calculation (-2+1)/(1+3) corresponds to (y1-y2)/(x2+x1)? No. y1=2, y2=1. x1=-1, x2=3. Rise = 1-2 = -1. Run = 3-(-1) = 4. Slope = -1/4. The equation shown is (-2+1)/(1+3). Numerator: -2+1 = -1. Denominator: 1+3 = 4. Result -1/4. This matches the correct slope value, but the expression (-2+1)/(1+3) does not represent the standard rise-over-run formula (y2-y1)/(x2-x1) or (y1-y2)/(x1-x2). It looks like (y1-2? no) ... it looks like (y1 - y2) / (x2 + x1) if y2=1, x2=3, x1=-1? No. It seems to be (y1 - 2) / (x2 + 1)? No. It is likely a garbled attempt at (1-2)/(3-(-1)) where signs were messed up but coincidentally yielded the right number, or it's just wrong algebraic representation. Regardless, the text says 'Rise over run' but the formula shown is not the standard definition and is confusing/wrong in structure. More critically, step 2 claims 'Point-slope form through (-1, 2)' but shows `2 - (x + 1)/4`. Point slope is `y - y1 = m(x - x1)`. So `y - 2 = -1/4(x - (-1))` => `y = 2 - 1/4(x+1)`. The equation shown is `2 - (x+1)/4 = 7/4 - x/4`. This is an identity, not the derivation of the line equation. It simplifies the RHS of the point-slope form. It doesn't explicitly state `y = ...`. However, the final answer is correct. The main issue is Step 1's formula is nonsensical as a general method, even if the numbers work out. A student would learn to calculate slope as (y1-2)/(x2+1) or something similar? No, it's just bad notation. But is it an error? Yes, the formula for slope is not (-2+1)/(1+3). That is specific to these numbers and written in a confusing way. It should be (1-2)/(3-(-1)).gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The slope calculation in line 1 is incorrect; it computes (y2-y1)/(x2+x1) instead of (y2-y1)/(x2-x1), resulting in -1/4 instead of the correct slopegpt-oss:20b: pass 2026-10-16
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.