To determine which set of ordered pairs does not represent a direct variation, we need to check if each set of pairs can be expressed in the form \(y = kx\) for some constant \(k\). This means the ratio \(y/x\) must be constant for all pairs in a set, except at the point where \(x = 0\).
Let's analyze each set:
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Set 1: (10, 2); (15, 3); (20, 4)
- For (10, 2): \( \frac{2}{10} = 0.2 \)
- For (15, 3): \( \frac{3}{15} = 0.2 \)
- For (20, 4): \( \frac{4}{20} = 0.2 \)
All ratios are the same (\(k = 0.2\)), so this set represents direct variation.
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Set 2: (0, 0); (-2, 4); (3, -6)
- For (0, 0): The ratio is undefined (or could be considered 0).
- For (-2, 4): \( \frac{4}{-2} = -2 \)
- For (3, -6): \( \frac{-6}{3} = -2 \)
The two non-zero pairs provide the same ratio (\(k = -2\)). Direct variation is valid since \( (0,0) \) is included.
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Set 3: (0, 0); (1, 3); (2, 4)
- For (0, 0): The ratio is again defined as 0.
- For (1, 3): \( \frac{3}{1} = 3 \)
- For (2, 4): \( \frac{4}{2} = 2 \)
The ratios differ (3 and 2), so this set does not represent direct variation.
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Set 4: (1, 1); (-2, -2); (3, 3)
- For (1, 1): \( \frac{1}{1} = 1 \)
- For (-2, -2): \( \frac{-2}{-2} = 1 \)
- For (3, 3): \( \frac{3}{3} = 1 \)
All ratios are the same (\(k = 1\)), indicating direct variation.
From the analysis, the set of ordered pairs that is not a direct variation is:
(0, 0); (1, 3); (2, 4).
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