Answer:
Constant of proportionality: [tex]\frac{1}{3}[/tex]
Equation: [tex]y=\frac{1}{3}x[/tex]
Step-by-step explanation:
The constant of proportionality is the ratio between y and x, according to the following formula
constant of proportionality = k = y/x
Take the given pairs and replace the data.
1.
[tex]k=\frac{\frac{2}{3}}{2}[/tex]
[tex]k = \frac{1}{3}[/tex]
2.
[tex]k = \frac{1}{3}[/tex]
3.
[tex]k=\frac{\frac{10}{3}}{10}[/tex]
[tex]k = \frac{1}{3}[/tex]
4.
[tex]k = \frac{4}{12}[/tex]
[tex]k = \frac{1}{3}[/tex]
As you can see, the constant is the same for all the pairs.
constant of proportionality = [tex]\frac{1}{3}[/tex]
Now, in order to know the equation make a graph of the given data.
The graph is in the picture
Since it is a line, use the equation of the line in point slope form
[tex]y- { y }_{ 1 } = m \left( x- { x }_{ 1 } \right)[/tex]
where (x1, y1) = any pair of the data
m = slope = constant of proportionality
Replace the data in the equation
[tex]y-1 = \dfrac{ 1 }{ 3 } \left( x-3 \right)[/tex]
Use the distributive property to multiply [tex]\frac{1}{3}[/tex] by x - 3
[tex]y-1=\frac{1}{3}x-1[/tex]
Add 1 to both sides
[tex]y = \dfrac{ 1 }{ 3 } x-1+1[/tex]
Add −1 and 1 to get 0
[tex]y=\frac{1}{3}x[/tex]
or
[tex]y=\frac{x}{3}[/tex]
