Position yourself in front of the outlet hole. Hold the carpenter's square up around the hole and mark a square larger than the hole on the sheetrock using a pencil. Make the marks large enough that they will go over at least halfway onto one stud, and if the original hole is large enough, over two studs in the wall.
Cut a hole in the sheetrock along the marked lines using the sheetrock saw, making sure the cut is as straight and square as possible. Cut the vertical portions of the hole that are positioned over half of the studs using the utility knife, as this will ensure that the patch has something to screw into, keeping it strong in the process.
Cut the 1-by-3 lumber to a size that's 2 to 4 inches larger than the vertical hole in the sheetrock using the saw. Slide the lumber into the hole in the wall so that it's behind the sheetrock. Position the lumber so that it's either in the center of the hole if there is a stud exposed on both sides, or on just one side if there is no other stud exposed. Screw the lumber to the sheetrock using the drill and the 3-inch sheetrock screws.
Cut out a piece of the sheetrock scrap that is the same size as the hole in the wall using the carpenter's square as a guide and the razor blade. Install the new sheetrock patch panel into the wall. Drill it to the lumber support and the studs using the drill and 3-inch sheetrock screws. Apply the fiberglass tape along the edges of the patch so that the tape goes halfway on the original sheetrock and halfway on the patch panel.
Apply joint compound across the fiberglass tape and the patch panel using the 9-inch putty knife, working the joint compound to form as straight and flat a surface as possible. Let the joint compound cure for 24 hours. Sand the joint compound with the fine sandpaper. Apply a second coat of joint compound using the 9-inch putty knife, allow it to cure for 24 hours, then sand again until the panel is flat. Finish the panel with one last pass of joint compound applied with the 12-inch putty knife, wait 24 hours, then sand if necessary so that the wall is ready for paint.