Two-column proof:

Step 2 can be split into two smaller steps: add ⅔z to each side (addition PoE); and subtract 1 from each side (subtraction PoE). Similarly, step 3 can be split into two smaller steps: multiply each side by 3 (multiplication PoE) then divide both sides by 2 (division PoE).
The reason in step 4 is also known as the symmetric property of equality.