To solve the equation
\[ -\frac{1}{14} = -\frac{1}{7} + z, \]
we need to isolate \( z \).
First, we can add \(\frac{1}{7}\) to both sides of the equation:
\[ -\frac{1}{14} + \frac{1}{7} = z. \]
Next, we need to find a common denominator to combine the fractions on the left side. The common denominator between 14 and 7 is 14. We can rewrite \(\frac{1}{7}\) as follows:
\[ \frac{1}{7} = \frac{2}{14}. \]
Now substitute this back into the equation:
\[ -\frac{1}{14} + \frac{2}{14} = z. \]
Now we combine the fractions:
\[ -\frac{1}{14} + \frac{2}{14} = \frac{-1 + 2}{14} = \frac{1}{14}. \]
So, we find that
\[ z = \frac{1}{14}. \]
Checking the solution:
We need to substitute \( z = \frac{1}{14} \) back into the original equation to verify it:
Starting with
\[ -\frac{1}{14} = -\frac{1}{7} + z, \]
this becomes
\[ -\frac{1}{14} = -\frac{1}{7} + \frac{1}{14}. \]
Now, rewrite \(-\frac{1}{7}\) as \(-\frac{2}{14}\):
\[ -\frac{1}{14} = -\frac{2}{14} + \frac{1}{14}. \]
Now combine the right side:
\[ -\frac{2}{14} + \frac{1}{14} = \frac{-2 + 1}{14} = \frac{-1}{14}. \]
Since both sides of the equation are equal, the solution is confirmed.
Thus, the final answer is
\[ \boxed{\frac{1}{14}}. \]