Answer:
Option D. ![y=\frac{1}{3}x](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B1%7D%7B3%7Dx)
Step-by-step explanation:
we know that
A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form
or ![y=kx](https://tex.z-dn.net/?f=y%3Dkx)
In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin
<u><em>Verify each case</em></u>
case A) we have
![y=x+\frac{1}{3}](https://tex.z-dn.net/?f=y%3Dx%2B%5Cfrac%7B1%7D%7B3%7D)
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
![y=0+\frac{1}{3}=\frac{1}{3}](https://tex.z-dn.net/?f=y%3D0%2B%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B1%7D%7B3%7D)
so
The line not passes through the origin
therefore
The equation A not represent a proportional relationship
case B) we have
![y=1-\frac{1}{3}x](https://tex.z-dn.net/?f=y%3D1-%5Cfrac%7B1%7D%7B3%7Dx)
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
![y=1-\frac{1}{3}(0)=1](https://tex.z-dn.net/?f=y%3D1-%5Cfrac%7B1%7D%7B3%7D%280%29%3D1)
so
The line not passes through the origin
therefore
The equation B not represent a proportional relationship
case C) we have
![y=3x+\frac{1}{3}](https://tex.z-dn.net/?f=y%3D3x%2B%5Cfrac%7B1%7D%7B3%7D)
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
![y=3(0)+\frac{1}{3}=\frac{1}{3}](https://tex.z-dn.net/?f=y%3D3%280%29%2B%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B1%7D%7B3%7D)
so
The line not passes through the origin
therefore
The equation C not represent a proportional relationship
case D) we have
![y=\frac{1}{3}x](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B1%7D%7B3%7Dx)
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
![y=\frac{1}{3}(0)=0](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B1%7D%7B3%7D%280%29%3D0)
so
The line passes through the origin
therefore
The equation D represent a proportional relationship