A function 𝑓(𝑥) has the property that, for all positive 𝑥, 3𝑓(𝑥)+7𝑓(2016𝑥)=2𝑥.What is the value of 𝑓(8)?

To solve this problem, we need to find a pattern or relationship that can help us determine the value of f(8).

1. Let's start by setting x = 1/2016 in the given equation. We get:

   3f(1/2016) + 7f(1) = 2/2016

2. Now, let's set x = 1 in the given equation. We get:

   3f(1) + 7f(2016) = 2

3. From the first equation, we can express f(1) as:

   f(1) = (2/2016 - 3f(1/2016)) / 7

4. Substitute f(1) from equation 3 into equation 2:

   3((2/2016 - 3f(1/2016)) / 7) + 7f(2016) = 2

5. Simplify the equation to find the value of f(2016):

   f(2016) = (2 - 3(2/2016 - 3f(1/2016)) / 7) / 7

6. Now, let's set x = 8 in the given equation. We get:

   3f(8) + 7f(20168) = 28

7. Substitute f(2016*8) from equation 5 into equation 6:

   3f(8) + 7((2 - 3(2/2016 - 3f(1/2016)) / 7) / 7) = 16

8. Solve the equation to find the value of f(8).

Please note that this is a complex problem and it may require additional steps or a different approach depending on the specific properties of the function f(x).