To solve the equation, let's first clarify the expressions you provided. It seems you're trying to manipulate the equation that involves fractions and variables. Let's break them down into two separate equations based on their appearance:

1. \( \frac{4}{3} - (x - \frac{1}{5}) = \frac{1}{5} - \frac{1}{6} \)
2. \( \frac{4}{3} - (x - \frac{1}{5}) = \frac{1}{5} - \frac{1}{64} \)

Let's solve these one by one.

### Equation 1:

\[
\frac{4}{3} - (x - \frac{1}{5}) = \frac{1}{5} - \frac{1}{6}
\]

First, simplify the right side:

\[
\frac{1}{5} - \frac{1}{6}
\]

Finding a common denominator (which is 30):

\[
\frac{6}{30} - \frac{5}{30} = \frac{1}{30}
\]

Now, our equation becomes:

\[
\frac{4}{3} - (x - \frac{1}{5}) = \frac{1}{30}
\]

Next, simplify the left side. Distributing the negative:

\[
\frac{4}{3} - x + \frac{1}{5} = \frac{1}{30}
\]

Now, we'll find a common denominator for \(\frac{4}{3}\) and \(\frac{1}{5}\). The least common multiple of 3 and 5 is 15:

\[
\frac{4}{3} = \frac{20}{15}, \quad \frac{1}{5} = \frac{3}{15}
\]

So,
\[
\frac{20}{15} + \frac{3}{15} = \frac{23}{15}
\]

Now our equation is:

\[
\frac{23}{15} - x = \frac{1}{30}
\]

Subtract \(\frac{23}{15}\) from both sides:

To combine the fractions, convert \(\frac{23}{15}\) to a fraction with a denominator of 30:

\[
\frac{23}{15} = \frac{46}{30}
\]

So the equation now reads:

\[
-x = \frac{1}{30} - \frac{46}{30}
\]

This simplifies to:

\[
-x = \frac{1 - 46}{30} = \frac{-45}{30} = -\frac{3}{2}
\]

Thus,

\[
x = \frac{3}{2}
\]

### Equation 2:

\[
\frac{4}{3} - (x - \frac{1}{5}) = \frac{1}{5} - \frac{1}{64}
\]

Again, we need to simplify the right side:

\[
\frac{1}{5} - \frac{1}{64}
\]

Find a common denominator, which is 320:

\[
\frac{64}{320} - \frac{5}{320} = \frac{64 - 5}{320} = \frac{59}{320}
\]

Now the left side becomes:

Using the previous left side:

\[
\frac{23}{15} - x = \frac{59}{320}
\]

Similar to before, we'll convert \(\frac{23}{15}\) to a fraction with a denominator of 320.

\[
\frac{23}{15} = \frac{23 \cdot 21.33}{15 \cdot 21.33} = \frac{326.3}{320}
\]

Now, rewrite the equation:

\[
\frac{326.3}{320} - x = \frac{59}{320}
\]

And rearranging gives:

\[
-x = \frac{59}{320} - \frac{326.3}{320}
\]

This results in:

\[
-x = \frac{59 - 326.3}{320} = \frac{-267.3}{320}
\]

Thus,

\[
x = \frac{267.3}{320}
\]

In conclusion, the solutions to the two equations are:

1. \( x = \frac{3}{2} \)
2. \( x = \frac{267.3}{320} \)

If you have further questions or need clarification on specifics, feel free to ask!