Eigenvalues and Eigenvectors

I'm struggling with finding the eigenvectors for this paritcular question . I understand how to compute eigenvalues and hence the corresponding eigenvectors but my problem seems to be with reducing the matrix for it to be more easily solvable .

My eigenvalue yields the 3x3 matrix ;

7 8 -7

-2 -1 2

2 4 -2

From which I must solve to find the eigenvector corresponding to the eigenvalue of -2 .

I tried to reduce the matrix to echelon form but reach a stage where I can solve the equation but the matrix hasn't been reduced fully to echelon form .

R1-3R3

R1 + R2

R3 + R1

R3 + R1

This yields the matrix -1 5 1

-2 -1 2

0 -6 0

From this I can see that y will be 0 which allows me to solve x and z .

This gives the eigenvector as (1,0,1) but this doesn't seem like I have completed the method correctly . If anyone could clarify what I've done as being correct or could help me with reducing to echelon form that would be great .

Thank you .