I've been doing something similar for years.

I used 1/2" Plywood, and then used 1/4-20 T-Nuts on the back side of the plywood, matching with the holes for the mounting plate, then used 1/4-20 x 1/2" Truss Head Machine Screws (truss head screws have a larger head than regular, so don't need a washer) to go into the T-Nuts.

Then I used the 2-1/2" (I think) lag bolts that came with the bracket to go into the Plywood and studs where they cleared the bracket on the outside, and/or 2-1/2" drywall screws in places that the lag bolts would get in the way of the mount.

This is a heavier articulating mount that pulls out, so perhaps I went a bit overkill, but I have used this same pieces in now 3 apartments, and just have to adjust where the screws go to match the studs and center the display accordingly.

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I have two boards like this, this picture is of a wider board I have that also has another mount to the right of my 49" display that is for a 27" display that is portrait for my desk setup, but the principles the same. You can see the countersunk holes from the drywall screws from a previous apartment.

Image:

https://app.box.com/s/i7zbb9adb5gsvbz93rk790g2xk5tjz6a