Open Audacity. Once it opens, choose "Project" from the main menu bar and choose "Import Audio." A file selection dialogue will appear. Navigate to the song you want to remove the backing instrumentals from and select it. Click "Open." A new track displaying the song's waveform will appear.
Choose "Project" and "Import Audio" again. In the file selection dialogue, navigate to the instrumentals-only version of the song. Select it and click "Open." Its waveform will appear on a new track beneath the first.
Align the first and second tracks as precisely as possible. Click the zoom-in button in the toolbar repeatedly until the waveforms appear as thin, wavy lines. The waveforms on each track will be nearly identical. Find a peak or valley that has exactly the same shape in both tracks.
Choose the "Time-Shift Tool" from the toolbar. It looks like a double-ended horizontal arrow. Carefully align the waveform shapes until they correspond as closely as possible by click-holding and dragging one of the tracks to the left or right as needed.
Click the disclosure triangle at the top of the first track's track header. From the pop-up menu, choose "Split Stereo Track." The track will split the left and right stereo outputs into separate mono tracks. Do the same for the second track. You will now have four separate mono tracks.
Delete the first and third mono tracks by clicking the "X" button. Fine-tune the alignment between the two remaining tracks following the procedure in Step 3. For phase cancellation to work, it is important that the tracks are aligned as closely as possible.
Click on the second track's track header to highlight it. Choose "Effects" from the menu bar and choose "Invert" from the menu. The second track's waveform will flip, becoming a mirror image of the original.
Click the "Play" button. Because the second track containing only the instrumentals is inverted, all sound information shared by both tracks will be canceled out. This will leave only that sound information that the tracks do not share. If the tracks have been aligned exactly, the only sound you will hear is the vocals.