Remove any carpeting that covers the subfloor by starting in one corner of the room and lifting the carpet off of its tack strip. Cut the carpet into 4-foot strips using a utility knife and roll the strips up individually for easy removal. Find the floor joists that run parallel on each side of the joist or joists that you're removing. Mark on the subfloor the center of the left and right parallel joists at each end and snap a chalk line across the entire length.
Remove any nails along the snapped lines using a hammer or a claw bar.
Adjust the circular saw's cutting depth to 1/16 inch deeper than the subfloor material; this will prevent you from cutting into the floor joist.. Plunge cut into the subfloor and follow the snapped lines. Slip the claw bar's flat end inside the cuts and pry up on the subfloor material to expose the joist that you're replacing.
Remove any nails that secure the joist into place. Check that there are no blocks or bracing to remove before lifting out the old joist. Grasp the joist and pull it straight up to remove.
Measure the joist's length and cut a new one out of lumber. Slide a hanger underneath the joist so that the tabs sit flush on the bottom and sides of the lumber. Flush the hanger to the end of the joist and hammer 8d nails through the allotted slots to secure the hanger to the joist. Slip the joist into the opening and secure the hanger to the rim joist with 8d nails.
Replace the cut subfloor with the sections that you cut out, or create new sections. Space 8d nails every 2 to 3 inches apart along all of the subfloor's outside edges. Space the nails 10 to 12 inches apart on the inside face of the subfloor material.