Lines and slopes
Problem 13.126 · easy
Find the slope of the line through \( \displaystyle (-3, 0) \) and \( \displaystyle (4, -3) \), say whether it is increasing, decreasing or horizontal, and give its equation in slope-intercept form.
- \[ \frac{-3 + 0}{3 + 4} = - \frac{3}{7} \]Rise over run.✓ Proved
- \[ - \frac{3 \left(x + 3\right)}{7} + 0 = - \frac{3 x}{7} - \frac{9}{7} \]Point-slope form through (-3, 0), then simplify.✓ Proved
- \[ \left. - \frac{3 x}{7} - \frac{9}{7} \right|_{\substack{ x=4 }} = -3 \]It passes through (4, -3) too.✓ Proved
Answer \( m = - \frac{3}{7}\ (\text{decreasing}),\quad y = - \frac{3 x}{7} - \frac{9}{7} \)
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 of the slope in step 1 is incorrect; it computes (y2-y1)/(x1+x2) instead of (y2-y1)/(x2-x1), resulting in -3/7 instead of the correct slope -3/7. Wait, let me re-check. Points are (-3, 0) and (4, -3). Slope m = (-3 - 0) / (4 - (-3)) = -3 / 7. The equation in step 1 is (-3 + 0)/(3 + 4). The numerator is -3 + 0 = -3. The denominator is 3 + 4 = 7. So the value is -3/7. The formula used is (y2 + y1) / (|x1| + x2)? No, it looks like it just added the x-coordinates magnitudes or something weird. x1 = -3, x2 = 4. x2 - x1 = 4 - (-3) = 7. The denominator in step 1 is 3 + 4 = 7. This is a coincidence because x1 is negative. If x1 were positive, this method would fail. The sentence says 'Rise over run'. Rise is -3. Run is 7. The calculation (-3+0)/(3+4) is numerically correct for these specific numbers but represents a fundamentally wrong formula for slope (it adds the absolute value of the first x-coordinate to the second, or assumes x1 is negative and flips sign). This is misleading/wrong method. A student would learn that slope is (y2+y1)/(x2+|x1|) or similar nonsense. It is an error in the method description/formula application.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The calculation of the slope in step 1 is incorrect; it computes (y2-y1)/(x1+x2) instead of (y2-y1)/(x2-x1), resulting in -3/7 instead of the correct slope -3/7. Wait, let me re-check. Points are (-3, 0) and (4, -3). Slope m = (-3 - 0) / (4 - (-3)) = -3 / 7. The equation in step 1 is (-3 + 0)/(3 + 4). The numerator is -3 + 0 = -3. The denominator is 3 + 4 = 7. So the value is -3/7. The formula used is (y2 + y1) / (|x1| + x2)? No, it looks like it just added the x-coordinates magnitudes or something weird. x1 = -3, x2 = 4. x2 - x1 = 4 - (-3) = 7. The denominator in step 1 is 3 + 4 = 7. This is a coincidence because x1 is negative. If x1 were positive, this method would fail. The sentence says 'Rise over run'. Rise is -3. Run is 7. The calculation (-3+0)/(3+4) is numerically correct for these specific numbers but represents a fundamentally wrong formula for slope (it adds the absolute value of the first x-coordinate to the second, or assumes x1 is negative and flips sign). This is misleading/wrong method. A student would learn that slope is (y2+y1)/(x2+|x1|) or similar nonsense. It is an error in the method description/formula application.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-14 — The calculation for the slope in step 1 is incorrect; the numerator should be the difference in y-coordinates (-3 - 0), not the sum (-3 + 0). Although the final numerical result (-3/7) is coincidentally correct, the reasoning 'Rise over run' is applied to an invalid expression.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The calculation of the slope in step 1 is incorrect; the numerator should be the difference in y-coordinates (-3 - 0), not the sum (-3 + 0). Although the final numerical result is coincidentally correct, the reasoning presented is mathematically false.gpt-oss:20b: fail (misleading) 2026-10-14 — Sentence 1 uses the expression (-3+0)/(3+4) instead of the correct (y₂−y₁)/(x₂−x₁)=(-3−0)/(4−(−3)). Although it evaluates to the right value, it misrepresents the formula and could confuse a student.
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.