To determine the number of solutions for the system of linear equations:
- \(2x + y = 6\) (Equation 1)
- \(4x + 2y = 8\) (Equation 2)
We can start by simplifying Equation 2. Notice that both coefficients in Equation 2 can be divided by 2:
\[ 4x + 2y = 8 \implies 2x + y = 4 \]
Now, we have the following simplified equations:
- \(2x + y = 6\)
- \(2x + y = 4\)
Next, we can observe that both equations represent lines in a two-dimensional space. By comparing them, we can see that:
- The line represented by \(2x + y = 6\) has a different intercept from the line represented by \(2x + y = 4\). In fact, both lines have the same slope but different y-intercepts, which indicates that they are parallel.
Since parallel lines do not intersect, we conclude that this system of equations has no solutions.
Thus, the number of solutions to the system of equations is:
\[ \boxed{0} \]
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