Question

Does the data in the table represent a direct variation or an inverse variation write an equation to model the data in the table x 6,8,12,20 y 9,12,18,30

Answer 1

direct variation

Explanation:

For direct variation k =
`(y)/(x)` ← k is the constant of variation

For inverse variation k = yx

Expressing the data as ordered pairs

(6, 9), (8, 12), (12, 18), (20, 30)

k =
`(9)/(6)` =
`(12)/(8)` =
`(18)/(12)` =
`(30)/(20)` =
`(3)/(2)` = 1.5 ← indicating direct variation

Equation is

y = kx = 1.5x

Answer 2

The data in the table represents a direct variation, as the ratio
`\( (y)/(x) \)` remains constant
`(\( (3)/(2) \))`. The equation for the direct variation is
`\( y = (3)/(2)x \).`

To determine whether the data represents direct variation or inverse variation, we can check if the ratio
`\( (y)/(x) \)` remains constant for all data points.

`\[ (y)/(x) = (9)/(6) = (12)/(8) = (18)/(12) = (30)/(20) = (3)/(2) \]`

Since the ratio
`\( (y)/(x) \)` is constant (equal to
`\( (3)/(2) \)`), the data represents a direct variation. In a direct variation, the ratio of y to x is a constant value.

Now, let's write the equation for direct variation. The general form is y = kx, where k is the constant of variation.

`\[ y = (3)/(2)x \]`

So, the equation that models the data in the table for direct variation is
`\( y = (3)/(2)x \).`