I’ve just finished the course in Audit mode for now. But see this post Eigenvalues & Eigenvectors quiz- how was the eigenvector derived? - #6 by Elemento.
As with others the eigenvectors are confusing initially. I’ve studied this stuff before but it’s been decades since I’ve done any math!
To answer your question, there are infinitely many eigenvectors, since they can be scalar multiples of each other. So for x = 2y, how do we know that y is 1? We don’t. We’re just choosing the simplest integer values that satisfy the equation. We could have chosen y = 2, in which case x = 4. Similar reasoning for y = -2x. Let x = 1, then y must be -2. But we could have said x = -1, in which case y = 2.
Btw, at various points through the course I also consulted other material, e.g., 3Blue1Brown’s Essence of Linear Algebra series on YouTube. I often find it helps if you can approach material from different angles. I also have an old Math A-level book that covers the topic. So I referred to that as well.