Compacted soil blocks water and air from reaching roots, which is why you're seeing those dead patches. I dealt with something similar in my backyard a couple years ago - the front had been packed down from years of foot traffic, and I started by just adding a thick layer of compost (like 2-3 inches) and letting it sit through a season. The organic matter slowly broke things up and helped water penetrate way better, plus the grass filled in as conditions improved. You can also try topdressing with compost mixed with sand, working it into the surface with a rake, which gives you some of the benefits of aeration without the equipment rental. If you go the manual fork route that was mentioned already, focus on getting it as deep as possible and really work it - it's exhausting but honestly worth it for smaller problem areas.