Solution: Average equation: $ \frac{(4z - 3) + (2z + 5) + (z + 1)}{3} = 6 $. Simplify numerator: $ 7z + 3 = 18 $. Subtract 3: $ 7z = 15 $. Divide: $ z = \frac{15}{7} $. \boxed{\dfrac{15}{7}}