Finding eigenlines & eigenvalues

[Roodles01]

In my example I have matrix A = (1 2)
. . . . . . . . . . . . . . . . . . . . . . (3 2)

Finding the eigenvalue through the method I understand & can get the result

i.e.
k = 4 & -1

I suspect my algebra is the shaky link, here, but to find the eigenline I find a bit more of a challenge.

OK I start by substituting the eigenvalue into the eigenvector equation;
Ax = kx
giving
(1 2) (x) = 4 (x)
(3 2) (y)...(y)


which gives rise to the following simultaneous equations
x + 2y = 4x
3x + 2y = 4y

Now the bit I don't get . . .
How do these both reduce to 3x - 2y = 0

I'm sure it's simple & I can't see the wood for the trees, but this is stupidly defeating me.. . . . Grrr!

Please could someone point me in the right direction.

 

[Roodles01]

Damn, it's so easy to suddenly see the solution.
OK I'm done

x + 2y = 4x
3x+ 2y =4y

goes to
-3x + 2y = 0
3x - 2y = 0

both reduce to
3x - 2y = 0

Thank you.

 

[Roodles01]

All the other examples I have had led to the simultaneous equations reducing to a single equation, from which I can find the eigenline equation for each eigenvalue (2 in this case).

The simultaneous equation I have now is;
3/4 x - 3/4 y = 0
-1/4 x - 1/4 y = 0

I am assuming I have the negative value wrong, but I may not, I suppose, so how can I get back to a single equation so I can find the eigenline equation for each eigenvalue.

 

[HallsofIvy]

Please tell us what the matrix in this problem is so that we can tell whether or not 2 really is an eigenvalue.

 

[Roodles01]

(11/4 -3/4)
(-1/4 9/4)

This is the original matrix. Thank you.

 

[Deveno]

All the other examples I have had led to the simultaneous equations reducing to a single equation, from which I can find the eigenline equation for each eigenvalue (2 in this case).

The simultaneous equation I have now is;
3/4 x - 3/4 y = 0
-1/4 x - 1/4 y = 0

I am assuming I have the negative value wrong, but I may not, I suppose, so how can I get back to a single equation so I can find the eigenline equation for each eigenvalue.

based on the matrix you provided, the second equation should be:

-(1/4)x + (1/4)y = 0.

thus BOTH equations lead to:

x = y.