Question

Which of the following sets of ordered pairs is not a direct variation?

(10, 2); (15, 3); (20, 4)
(0, 0); (-2, 4); (3, -6)
(0, 0); (1, 3); (2, 4)
(1, 1); (-2, -2); (3, 3)

Answer 1

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:

1. **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.

2. **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.

3. **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.

4. **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)**.