Linearly independent vectors

(Note: the **bold** letters are vectors.)

"Suppose that **v** and **w** are vectors which are not parallel (so are linearly independent) and the following vector equation holds for some scalars α and β.

**v** + α(**w** - **v**) = β(**v** + 1/2**w**).

Find α and β."

With my working out:

-> (1 − α − β) v + (α − β/2)w = 0

-> 1 − α − β = 0 = α − β/2 (by linear independence)

...and from this, I attempted to solve the equations simultaneously but did not get the right answer. (Which was, α = 1/3, β = 2/3)

I really don't understand how you could get that answer either. So, my only problem here is just solving the equations to get the final answer.