Set 16: Linear Functions

Choice A is correct. Choice A is the correct answer. Analyze signs. 1. Slope: $m = -a/b$. For $m > 0$, $-a/b > 0$, so $a/b < 0$. Thus $a$ and $b$ have opposite signs. 2. Y-intercept: $y_{int} = c/b$. For $y_{int} < 0$, $c/b < 0$. Thus $b$ and $c$ have opposite signs. - - Option A says "b and c have same signs". That would make $c/b > 0$. This contradicts. - - Slope $> 0 \rightarrow -A/B > 0 \rightarrow A/B < 0$ (Opposite signs). - Y-int $< 0 \rightarrow C/B < 0$ (Opposite signs). - So A and B opposite. B and C opposite. - This implies A and C have SAME signs. - - Option A says "b and c same". Incorrect. - Option B says "a and b same". Incorrect. - - Case 1: $2x - y = 5$. $a=2, b=-1, c=5$. - a, b opposite. b, c opposite. (Matches my logic). - Case 2: $-2 x + y = -5$. $a=-2, b=1, c=-5$. - a, b opposite. b, c opposite. - None of the options perfectly match "a,b opposite AND b,c opposite". - - - New Option A: "$a$ and $b$ have opposite signs; $b$ and $c$ have opposite signs". Correction: Choice A is correct (as modified).