To solve the given equation, we will use the trigonometric identities and properties to manipulate the expressions and simplify them. Let's break down the problem into steps:

Step 1: Understanding the given equation
We are given:
α - β = π/6
2sin(α) - cos(β) = √3sin(β)

Step 2: Rearranging the first equation
From the first equation, we can rearrange it to find the value of α in terms of β:
α = β + π/6

Step 3: Substituting α in terms of β in the second equation
Now, let's substitute the value of α from the rearranged equation into the second equation:
2sin(β + π/6) - cos(β) = √3sin(β)

Step 4: Applying trigonometric identities
Using the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B), we can rewrite the equation as:
2(sin(β)cos(π/6) + cos(β)sin(π/6)) - cos(β) = √3sin(β)

Since cos(π/6) = √3/2 and sin(π/6) = 1/2, we can simplify the equation further:
2(√3/2sin(β) + 1/2cos(β)) - cos(β) = √3sin(β)

Simplifying the equation:
√3sin(β) + cos(β) - cos(β) = √3sin(β)

Step 5: Simplifying the equation
From the equation above, we can see that the term cos(β) cancels out on both sides:
√3sin(β) = √3sin(β)

Step 6: Finalizing the solution
The equation is now balanced, indicating that the given equation holds true for any value of β. Therefore, there are infinitely many solutions for α and β that satisfy the given equation.

In conclusion, the equation 2sin(α) - cos(β) = √3sin(β) is satisfied when α - β = π/6. The solution shows that there are infinitely many values of α and β that satisfy the equation.