I am not sure what or how the prof approached this problem after step 6.
It'd be greatly appreciated if anyone can tell me what's going on here.
12 in your sixth arrow should this be
1/2? If this is a typo (either in your slides or this post) then the rest of the problem seems straightforward, multiply by 2 to get $2y(x)=y(x+1) +y(x-1)$ then $2y(x) = y(x) + y(x)$ and rearrange to get the 7th arrow. Then the problem is a matter of fitting an affine function to 2 points.