For each table, determine whether it shows a direct variation. If it does, write its direct variation equation.

| x | y | | x | y |
|---|---|---|---|---|---|
| 2 | 4 | Direct variation Equation: | 2 | 5 | Not direct variation |
| 4 | 8 | | 4 | 10 |
| 8 | 16 | | 5 | 17.5 |

Table 1 has a constant ratio of x y ​ = 2 , so it is a direct variation with equation y = 2 x .
Table 2 does not have a constant ratio of x y ​ , so it is not a direct variation.
Direct variation equation for Table 1: y = 2 x .
Table 2: Not a direct variation.

Explanation

Understanding Direct Variation We are given two tables and asked to determine if each represents a direct variation. A direct variation has the form y = k x , where k is the constant of variation. To check if a table represents a direct variation, we need to see if the ratio x y ​ is constant for all pairs of x and y in the table.

Analyzing Table 1 For the first table, we have the pairs (2, 4), (4, 8), and (8, 16). Let's calculate the ratio x y ​ for each pair:



For (2, 4): 2 4 ​ = 2
For (4, 8): 4 8 ​ = 2
For (8, 16): 8 16 ​ = 2

Since the ratio is constant and equal to 2, the first table represents a direct variation with k = 2 . Therefore, the equation is y = 2 x .

Analyzing Table 2 For the second table, we have the pairs (2, 5), (4, 10), and (5, 17.5). Let's calculate the ratio x y ​ for each pair:


For (2, 5): 2 5 ​ = 2.5
For (4, 10): 4 10 ​ = 2.5
For (5, 17.5): 5 17.5 ​ = 3.5

Since the ratio is not constant (2.5, 2.5, and 3.5), the second table does not represent a direct variation.

Conclusion In summary, the first table represents a direct variation with the equation y = 2 x , and the second table does not represent a direct variation.

Examples
Direct variation is a fundamental concept in many real-world scenarios. For instance, the distance you travel at a constant speed varies directly with the time you spend traveling. If you're driving at a steady 60 miles per hour, the equation d = 60 t represents this direct variation, where d is the distance and t is the time. Similarly, the cost of buying multiple items of the same price varies directly with the number of items. If each apple costs 0.75 , t h e t o t a l cos t c f or n a ppl es i s c = 0.75n$. Understanding direct variation helps in predicting outcomes and making informed decisions in everyday situations.

Answered by GinnyAnswer | 2025-07-08

Table 1 shows a direct variation with the equation y = 2 x , while Table 2 does not show a direct variation as the ratios are not constant.
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Answered by Anonymous | 2025-07-28