Use a utility knife to cut a shallow groove along both sides of the ceiling crack, then remove any debris from the cracked area.
Push on the piece of drywall to the left of the crack and see if there is any movement, even if it's very minimal. Do the same to the piece of drywall to the right of the crack. If the drywall moves, use the drywall screws and drill to secure the drywall to the stud. Put the screw as close to the crack as possible so that it'll grip the wood. Screws should be placed approximately every 12 inches along the crack.
Cover the cracked area with mesh drywall tape. Mesh tape is a better option over paper tape in this application because the back side of the mesh has adhesive, so it will stick to the ceiling without you having to embed it in joint compound.
Load your mud pan with joint compound.
Apply a very thin layer of compound over each side of the taped crack, by using the 10-inch drywall knife and running it over the surface. Make one more pass over the patch, this time down the center to smooth it all out. Allow this coat of compound to dry.
Scrape off any edges or chunky bits of compound over the crack by running your 10-inch putty knife over the patched area.
Apply a second coat of joint compound the same way you applied the first coat and allow it to dry.
Sand the patched area lightly with a fine-grit sanding sponge, taking caution to not sand the drywall, just the patched area. Drywall has a paper surface and sanding it can tear and damage the surface.
Check the area for smoothness by running your hand over the patched area and taking note of any high or low spots. Also, place the tip of the blade of the drywall knife up against the ceiling, which will show you any high or low spots in the patched area. Apply another coat of drywall compound if necessary to smooth and taper the area.
Prime and paint the ceiling. Follow the application instructions on the paint and primer.