To determine the number of solutions for the system of equations, we can analyze the given equations:
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\( 12x + 6y = 3 \)
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\( 4x + 2y = 1 \)
First, let's simplify both equations:
For the first equation:
\[ 12x + 6y = 3 \implies 2x + y = \frac{1}{2} \]
(we divide every term by 6)
For the second equation:
\[ 4x + 2y = 1 \implies 2x + y = \frac{1}{2} \]
(we divide every term by 2)
Now we can see that both simplified equations are:
\[ 2x + y = \frac{1}{2} \]
Since both equations are identical, they represent the same line in the coordinate plane. Therefore, there are infinitely many solutions to this system of equations, as any point on the line satisfies both equations.
In conclusion, the system of equations has infinitely many solutions.
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