Final answer:
The given system of equations has Option 3: infinite many solutions.
Step-by-step explanation:
To determine how many solutions the given system of equations has, we need to solve the equations and see if they intersect at one point (one solution), are parallel (no solution), or overlap (infinite solutions).
First, let's simplify each equation:
3(x-3) = 10x+2 becomes 3x-9 = 10x+2
-9x+3 = -3(3x-1) becomes -9x+3 = -9x+3
2(3x-1) = 6x-1 becomes 6x-2 = 6x-1
Now, we can solve these equations.
Simplifying the first equation, we have 3x-9 = 10x+2. Subtracting 3x from both sides gives -9 = 7x+2. Subtracting 2 from both sides gives -11 = 7x. Dividing both sides by 7 gives x = -11/7.
Substituting this value back into the other two equations, we find that all the variables cancel out and we are left with true statements.
Therefore, the system of equations has infinite many solutions.