Let's solve the equation step by step:
Starting with the equation:
\[ 6(3z - 2) + 4(z - 5) = 78 \]
First, distribute the \(6\) and \(4\):
\[ 6 \cdot 3z - 6 \cdot 2 + 4 \cdot z - 4 \cdot 5 = 78 \] \[ 18z - 12 + 4z - 20 = 78 \]
Next, combine like terms:
\[ (18z + 4z) + (-12 - 20) = 78 \] \[ 22z - 32 = 78 \]
Now, add \(32\) to both sides:
\[ 22z - 32 + 32 = 78 + 32 \] \[ 22z = 110 \]
Next, divide both sides by \(22\):
\[ z = \frac{110}{22} \] \[ z = 5 \]
Thus, the solution for \(z\) is:
\[ \boxed{5} \]