To determine how many solutions the system of equations has, we can first analyze the two given equations:
- \( 12x + 6y = 3 \) (Let's call this Equation 1)
- \( 4x + 2y = 1 \) (Let's call this Equation 2)
Next, we can manipulate these equations to see if they are equivalent or if they represent different lines.
Let's simplify Equation 2:
- Divide the entire equation by 2: \[ 2x + y = \frac{1}{2} \]
Now let's also simplify Equation 1:
- Divide the entire equation by 3: \[ 4x + 2y = 1 \]
After simplification, we can see that both Equation 1 and Equation 2 simplify to the same equation:
- \( 4x + 2y = 1 \)
This means the two equations are equivalent, and they represent the same line in the plane.
Since both equations are the same, there are infinitely many solutions to this system of equations, as every point on the line represented by \(4x + 2y = 1\) is a solution.
Therefore, the correct answer is:
Infinitely many solutions.
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