Option A is the relationship which shows a direct variation.
Step-by-step explanation:
The direct variation is a relationship between two variables in which one is the multiple of the other. It is given by the relation
![\frac{y}{x}=k](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%3Dk)
Option A:
For
and
,
![\frac{y}{x}=\frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%3D%5Cfrac%7B1%7D%7B2%7D)
For
and
,
![\frac{y}{x}=\frac{2}{4}= \frac{1}{2}](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%3D%5Cfrac%7B2%7D%7B4%7D%3D%20%5Cfrac%7B1%7D%7B2%7D)
Since, the constant k is equal for all the values of x and y in the table, this relationship is a direct variation.
Option B:
For
and
,
![\frac{y}{x}=\frac{4}{2}= 2](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%3D%5Cfrac%7B4%7D%7B2%7D%3D%202)
For
and
,
![\frac{y}{x}=\frac{16}{4} =4](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%3D%5Cfrac%7B16%7D%7B4%7D%20%20%3D4)
Since, the values of constant k is not equal for all the values of x and y in the table, this relationship is not a direct variation.
Option C:
For
and ![y=3](https://tex.z-dn.net/?f=y%3D3)
![\frac{y}{x} =\frac{3}{1} =3](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%20%3D%5Cfrac%7B3%7D%7B1%7D%20%3D3)
For
and ![y=5](https://tex.z-dn.net/?f=y%3D5)
![\frac{y}{x} =\frac{5}{2}](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%20%3D%5Cfrac%7B5%7D%7B2%7D)
Since, the values of constant k is not equal for all the values of x and y in the table, this relationship is not a direct variation.
Option D:
For
and ![y=-1](https://tex.z-dn.net/?f=y%3D-1)
![\frac{y}{x} =\frac{-1}{1} =-1](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%20%3D%5Cfrac%7B-1%7D%7B1%7D%20%3D-1)
For
and ![y=2](https://tex.z-dn.net/?f=y%3D2)
![\frac{y}{x} =\frac{2}{2} =1](https://tex.z-dn.net/?f=%5Cfrac%7By%7D%7Bx%7D%20%3D%5Cfrac%7B2%7D%7B2%7D%20%3D1)
Since, the values of constant k is not equal for all the values of x and y in the table, this relationship is not a direct variation.
Thus, Option A is the relationship which shows direct variation.