To determine the number of solutions to the system of equations:
- \(4x + 8y = 12\)
- \(3x + 6y = 9\)
we can start by simplifying these equations.
Equation 1: Divide the entire equation by 4: \[ x + 2y = 3 \quad \text{(Equation 1')} \]
Equation 2: Divide the entire equation by 3: \[ x + 2y = 3 \quad \text{(Equation 2')} \]
Now we have two equations:
- \(x + 2y = 3\) (from Equation 1')
- \(x + 2y = 3\) (from Equation 2')
Since both equations are identical, they represent the same line in the coordinate plane. This means that there are infinitely many solutions, as any point on the line defined by \(x + 2y = 3\) is a solution to the system.
Thus, the number of solutions to the system is infinite.