To solve the equation 3x(x-1)-x+1=0, we can start by simplifying the expression on the left side:

3x(x-1) - x + 1 = 0
3x^2 - 3x - x + 1 = 0
3x^2 - 4x + 1 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most efficient method.

The equation can be factored as follows:

(3x - 1)(x - 1) = 0

Now, we can set each factor equal to zero and solve for x:

3x - 1 = 0 or x - 1 = 0

Solving the first equation:

3x = 1
x = 1/3

Solving the second equation:

x = 1

Therefore, the solutions to the equation 3x(x-1)-x+1=0 are x = 1/3 and x = 1.