Can you help me understand why my answer is wrong (system of differential equations, solved via eigenvectors)?

This is homework, so please do not directly give me any answers!

I have a system of differential equations which must be solved using eigenvectors (I know there are other ways, but I have to use eigenvectors):
dx/dt = 4x + y
dy/dt = 4y

So my matrix A is:
[4 1]
[0 4]

Fluther formatting is making this hard, but just pretend that is one set of vertical square brackets enclosing all four numbers.

I get a repeated root of 4 for the eigenvalue. I’m confident this part is right.

Then I have to find the eigenvector, and I think this part might be where I am screwing up, although I can’t figure out how.

[4 1] * [x] = 4 * [x]
[0 4]...[y].........[y]

This translates to the two equations:

4x + y = 4x
4y = 4y

The latter equation gives us no information, but 4x’s in the first cancel and you get y=0, x is free. So for my eigenvector I choose:

[1]
[0]

The formula then to find x(t) and y(t) is:

[x(t)] = (at + b) * e^(eigenvalue*t) * eigenvector
[y(t)]

where a and b are constants.

So we have x(t) = (at + b) * e^(4t) and y(t) = 0.

This very well parallels a similar example problem the prof did in class, but the thing is, I know this answer isn’t right. I know because when I try to check my answer by differentiating x with respect to t, my answer is not equivalent to 4x + y.

dx/dt = a*e^(4t) + 4(at+b)*e^(4t)
4x + y = 4(at+b)*e^(4t)

I can see that y should equal a*e^(4t) to make this equation true (and this would be consistent with the equation for dy/dt as well), but I don’t know how that can be when I get 0 in the y entry of the eigenvector.

So greatly appreciate it if anyone is able to help!