To determine the number of solutions to the system of equations:
- \( 2x + y = 6 \)
- \( 4x + 2y = 8 \)
We can simplify the second equation. Notice that the second equation can be written as:
\[ 4x + 2y = 8 \implies 2x + y = 4 \]
Now we have the two equations:
- \( 2x + y = 6 \)
- \( 2x + y = 4 \)
Since both equations represent straight lines, we can see if they intersect by comparing them:
\[ 2x + y = 6 \quad (1) \] \[ 2x + y = 4 \quad (2) \]
The left-hand sides of both equations are exactly the same, but the right-hand sides are different. This means the two lines are parallel and will never intersect.
Thus, the system of equations has no solutions.
The correct answer is:
- 0