In the $xy$-plane, line $k$ does not pass through the origin. Which of the following statements, each taken alone, is individually sufficient to conclude that the slope of line $k$ is negative? Select all that apply.
For a non-vertical line with $x$-intercept $p$ and $y$-intercept $q$ (both non-zero since the line avoids the origin), slope $= -\dfrac{q}{p}$.
Statement A — $p = 2q$: Slope $= -\dfrac{q}{2q} = -\dfrac{1}{2} < 0$. Always negative (as long as $q \neq 0$). Sufficient. ✓
Statement B — $pq > 0$: Then $p$ and $q$ have the same sign. Slope $= -\dfrac{q}{p} = -({\rm same\ sign\ ratio}) < 0$. Always negative. Sufficient. ✓
Statement C — $(a-r)(b-s) < 0$: Slope $= \dfrac{b-s}{a-r}$. Since $(a-r)(b-s) < 0$, numerator and denominator have opposite signs, so slope $< 0$. Sufficient. ✓
Statement D — $y$-intercept $= 0$: This means the line passes through the origin, contradicting the given condition. Invalid / not applicable. ✗
Statements A, B, and C are each individually sufficient.
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