Lines and slopes
Problem 13.140 · easy
Find the slope of the line through \( \displaystyle (0, 3) \) and \( \displaystyle (4, -2) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-3 - 2}{0 + 4} = - \frac{5}{4} \]Rise over run.✓ Proved
- \[ 3 - \frac{5 \left(x + 0\right)}{4} = 3 - \frac{5 x}{4} \]Point-slope form through (0, 3), then simplify.✓ Proved
- \[ \left. 3 - \frac{5 x}{4} \right|_{\substack{ x=4 }} = -2 \]It passes through (4, -2) too.✓ Proved
Answer \( m = - \frac{5}{4}\ (\text{decreasing}),\quad y = 3 - \frac{5 x}{4} \)
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: fail (error) — Step 2 omits the dependent variable y in the point‑slope expression; it should read y - 3 = (-5/4)(x - 0) leading to y = 3 - 5x/4.qwen3.6:27b-mlx: pass — The solution correctly calculates the slope, identifies the line as decreasing, and provides the correct equation in slope-intercept form. The algebraic steps are verified and the logic is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly calculates the slope, identifies the line as decreasing, and provides the correct equation in slope-intercept form. The algebraic steps are verified and the logic is sound.gpt-oss:20b: fail (error) 2026-10-17 — Step 2 omits the dependent variable y in the point‑slope expression; it should read y - 3 = (-5/4)(x - 0) leading to y = 3 - 5x/4.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The calculation for the slope in step 1 is incorrect; it computes (y1 - y2)/(x1 + x2) instead of (y2 - y1)/(x2 - x1). The correct slope is -5/4, but the derivation shown yields -5/4 only by coincidence of signs if interpreted loosely, but strictly (-3-2)/(0+4) = -5/4 is arithmetically correct for the numbers written, yet the formula 'Rise over run' implies (y2-y1)/(x2-x1). Let's re-read carefully. Point 1: (0,3), Point 2: (4,-2). Rise = -2 - 3 = -5. Run = 4 - 0 = 4. Slope = -5/4. The solution writes (-3 - 2)/(0 + 4). This is (-5)/4 = -5/4. Wait, where did -3 come from? The y-coordinate is 3. The solution uses -3. This is a sign error in the setup of the numerator. It should be (-2 - 3) or (3 - (-2)) depending on order, but using -3 suggests the point was (0, -3) or the subtraction was flipped incorrectly. Actually, if it meant y1 - y2 with y1=3, it should be 3 - (-2) = 5. If it meant y2 - y1 with y2=-2, it should be -2 - 3 = -5. The expression (-3 - 2) implies y1=-3. This is a factual error in the setup of the calculation.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The calculation of the slope in step 1 is incorrect; it computes (-3 - 2)/(0 + 4) = -5/4, but the correct rise is -2 - 3 = -5 and run is 4 - 0 = 4, so the arithmetic in the numerator uses the wrong sign for the first y-coordinate (using -3 instead of 3) and the wrong sign for the second y-coordinate (using 2 instead of -2) relative to the standard formula (y2-y1)/(x2-x1). Wait, let's re-read carefully. The points are (0,3) and (4,-2). Slope m = (-2 - 3) / (4 - 0) = -5/4. The solution writes (-3 - 2)/(0 + 4). This evaluates to -5/4. However, the terms -3 and 2 do not correspond to the coordinates of the points in any standard order. It looks like it might be trying to do (y1 - y2)/(x1 - x2) but got the signs wrong? If it were (3 - (-2))/(0 - 4) = 5/-4 = -5/4. Or (-2 - 3)/(4 - 0) = -5/4. The expression (-3 - 2) suggests using y1=-3 and y2=2, which are not the given points. This is a 'magic number' error where the result is correct by coincidence or bad algebra, but the setup is wrong.gpt-oss:20b: pass 2026-10-14
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.