To determine the number of solutions to the system of equations:
- \( 4x + 8y = 12 \)
- \( 3x + 6y = 9 \)
we will rearrange both equations to check their relationship.
First, let's simplify each equation.
Starting with the first equation:
\[ 4x + 8y = 12 \]
Dividing the entire equation by 4:
\[ x + 2y = 3 \]
Next, for the second equation:
\[ 3x + 6y = 9 \]
Dividing the entire equation by 3:
\[ x + 2y = 3 \]
Now, we can see that both equations simplify to the same equation:
\[ x + 2y = 3 \]
Since both equations are identical, they represent the same line in a 2D coordinate system. Therefore, there are infinitely many solutions, as any point on this line will satisfy both equations.
Thus, the correct response is:
infinite