To find the general solution for the equation \(8(x - y) = 5(x - 3y) + 1\), we will first simplify and solve for \(x\) in terms of \(y\).

Starting with the given equation:
\[8(x - y) = 5(x - 3y) + 1\]

First, distribute the constants on both sides:
\[8x - 8y = 5x - 15y + 1\]

Next, move all terms involving \(x\) and \(y\) to one side of the equation:
\[8x - 5x - 8y + 15y = 1\]

Simplify the equation:
\[3x + 7y = 1\]

Now, solve for \(x\) in terms of \(y\):
\[3x = 1 - 7y\]
\[x = \frac{1 - 7y}{3}\]

Thus, the general solution for \(x\) in terms of \(y\) is:
\[x = \frac{1 - 7y}{3}\]

So, the general solution for the equation \(8(x - y) = 5(x - 3y) + 1\) is:
\[
\begin{cases}
x = \frac{1 - 7y}{3} \\
y = y
\end{cases}
\]
where \(y\) can be any real number.