Failed installation of CUDA - nvcc won't work and can't set PATH

I’m trying to install a cudatoolkit version compatible with a github repo I’m trying to work with that uses pytorch 1.11.0.

After installing either cudatoolkit=10.2 or 11.3 along with torch, torchvision, and torchaudio as specified on the pytorch website, there is no folder in /usr/local/ or anywhere else, so I don’t see anywhere for my environmental variables to point to to make nvcc work. Is there a different way to get this cudatoolkit to work and for nvcc --version to work?

For reference:

Ubuntu 22.04.3 LTS
NVIDIA-SMI 535.129.03 Driver Version: 535.129.03 CUDA Version: 12.2 
NVIDIA GeForce GTX 1070