Eigenvectors help :')

You've done everything correct, except the final bit.

When you sub the eigenvalues into the matrix
A-lambda*I
the matrix is singular, so the eigenvectors cannot be determined uniquely. However, you can assume their magnitude is also unity (for instance, which gives you an extra degree of freedom to determine them.

So for the final step, you get (from either equation)
x_1 = 2/3 * x_2
so the eigenvector is something like
v = a*[2/3, 1]
where a is a scalar.
Requiring the eigenvector to have unity magnituce allows you to get a value for a.

If you think its a bit weird, go back to the definition of an eigenvector / eigenvalue, .i.e

A*v = lambda*v

i.e the matrix product simply scales the eigenvector. If you double the eigenvector, this still holds as
A*2*v = 2*lambda*v
so the eigenvector cannot be determined uniquely. Its direction can, but not its magnitude.

Ahh i havent come across any of this - just the very basics i need to know i.e. how to do it rather than definitions and concept/theory.

Those 2 equations are correct (at least for what i need to understand - on mark scheme).

What ive gathered from what you said is that i take one of the equations and rearrange so that it becomes x_1 = x_2

-3x_1 = -2x_2

then you use inspection? so x_1 = 2 and x_2 = 3

So one possible eigenvector is (2 3) - written as 3 under 2

?

Apparently this is a further maths topic (which i didnt do) - ive looked online and read up on notes but im still confused!

How do you find eigenvectors?

So i have this matrix and i found the eigenvalues then subbed in 4 to find eigenvalues (yes i didnt include the working bc i know its right and irrelevant to the question). I get these 2 equations and this is where i get stuck - how do i solve them to find eigenvectors?

Normal simultaneous equations doesnt work and apparently you find the values together but i dont understand how

You've got to remember that your matrix: $\begin{pmatrix} a_{11} - \lambda & a_{12} \\ a_{21} & a_{22} - \lambda \end{pmatrix}$ will only have rank 1. So you don't expect a unique solution but a line of solutions.

The rank of a matrix isn't on the syllabus for A-Level FM, so you might want to edit your post to clarify it for the OP.

As i mentioned in my original post this is not for further maths ( i am aware it is a topic in fm). And no i understand what @Ryanzmw has put because that is what i was taught.