How does one use Linear Independent Vectors to satisfy a vector equation with Alpha and beta constants

Hey Everyone,

I've got my Uni exams next week and I'm totally lost with this idea revolving around Linear Independence, could someone point me towards an easy explanation or better yet just post the explanation.

On top of this I've got the following question from a past paper which I'd like an step by step explanation too if possible or even just an initial hint once Linear Independence is explained so that i could go ahead and solve it myself!

"Suppiose x and y are linearly independent vectors satisfying the following vector equation:

-4x+alpha(x+y) = beta(x-3y)

Solve for alpha and beta"

Any help will be greatly appreciated!!!!

-4x + α(x + y) = β(x - 3y)
-4x + αx + αy = βx - 3βy
-4x + αx + αy - βx + 3βy = 0
(α - β - 4)x + (α + 3β)y = 0

Now we use that fact that if x and y are linearly independent, then the only way a linear combination of x and y can equal zero is if their coefficients are equal to zero (that's really all you need to know about linear independence.) In other words, we have:

α - β - 4 = 0
α + 3β = 0

Subtract the second from the first:

-4β - 4 = 0
β = -1

α + 3β = 0
α + 3(-1) = 0
α = 3

Done!