Let's analyze the steps in the equation to determine which statement is true.
Starting with the original equation: \[ 3x - 9x + 1 = 2(-3x + 1) - 1 \]
Combine like terms on the left side: \[ -6x + 1 = 2(-3x + 1) - 1 \]
Distributing on the right side: \[ -6x + 1 = -6x + 2 - 1 \] \[ -6x + 1 = -6x + 1 \]
At this point, we have: \[ -6x + 1 = -6x + 1 \]
This is a true statement, as both sides of the equation are equal.
Therefore, the true statement is that \(-6x + 1 = -6x + 1\) holds true.