Answer:
Option D. [tex]y=\frac{1}{3}x[/tex]
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 [tex]y/x=k[/tex] or [tex]y=kx[/tex]
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
Verify each case
case A) we have
[tex]y=x+\frac{1}{3}[/tex]
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
[tex]y=0+\frac{1}{3}=\frac{1}{3}[/tex]
so
The line not passes through the origin
therefore
The equation A not represent a proportional relationship
case B) we have
[tex]y=1-\frac{1}{3}x[/tex]
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
[tex]y=1-\frac{1}{3}(0)=1[/tex]
so
The line not passes through the origin
therefore
The equation B not represent a proportional relationship
case C) we have
[tex]y=3x+\frac{1}{3}[/tex]
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
[tex]y=3(0)+\frac{1}{3}=\frac{1}{3}[/tex]
so
The line not passes through the origin
therefore
The equation C not represent a proportional relationship
case D) we have
[tex]y=\frac{1}{3}x[/tex]
Remember that
the line must pass through the origin
so
For x=0, y=0
In this case
For x=0
[tex]y=\frac{1}{3}(0)=0[/tex]
so
The line passes through the origin
therefore
The equation D represent a proportional relationship
