You can use matrices.
If X is the vector (x,y) then the equation is
where A is the matrix with -1 and 6 in the first row, and -1 and 4 in the second row
This matrix is diagonalisable since its eigenvalues are 1 and 2
The diagonal matrix is D with 1 ; 0 in the first row and 0 ; 2 in the second row
Then you can find the matrix P such that
Then the initial equation becomes which can be easily solved
Then you can find X by