To solve for a variable, we simply need to isolate the variable and simplify all other terms. We can do this by performing the opposite operations to both sides.
In this case, we also need to remember that if our fractions have different denominators, we multiply the numerator and the denominator by whatever value makes them have equal denominators. That is 3, here.
z-4/9-1/3=5/9
+4/9 +4/9
z-1/3=9/9
z-3/9=9/9
+3/9+3/9
z=12/9
Using the math above, we can see that z=12/9 or 1 1/3.