Question

For the data in the table, tell whether y varies directly with x . If it does, write an equation for the direct variation. x 0,1,2,3 y 0,4,8,12

a no
b yes,y = 4x
c yes y = 0.25x
d yes y = x+4

Answer 1

Yes, y varies directly with x. The equation for direct variation is: y = 4x

Determining Direct Variation from Data

To determine whether y varies directly with x from a set of data points, we can calculate the ratio of y to x for each data point. If the ratio is constant for all data points, then y varies directly with x.

Example

Consider the following data set:

x y

0 0

1 4

2 8

3 12

Two variables, x and y, are said to vary directly if there is a constant proportion between them. This means that as x increases, y increases proportionally, and as x decreases, y decreases proportionally. The constant proportion between x and y is represented by the equation:

y = kx

where k is the constant of proportionality.

As we can see, the ratio of y to x is constant for all data points, equal to 4. Therefore, y varies directly with x.

Equation for Direct Variation

In the case where y varies directly with x, the equation for direct variation can be written as:

y = kx

where k is the constant of proportionality. In this case, k = 4, so the equation for direct variation is: y = 4x

Answer 2

yes, y varies directly with x.

y=4x is an equation of the Direct Variation

Explanation:

x y

0 0

1 4

2 8

3 12

From the above data:

we can say that y varies directly with x.

Direct variation states that if y is expressed as the product of some constant number k and x.

i.e y=kx

Now, let any points from above data x= 2 and y= 8 to calculate the value of k,

8=2k

⇒
`k=(8)/(2) =4`

Therefore, the equation for the direct variation is, y=4x