To determine the number of solutions to the system of linear equations:
- \(2x + y = 6\)
- \(4x + 2y = 8\)
let's rewrite the second equation.
Notice that the second equation can be simplified:
\[ 4x + 2y = 8 \]
Dividing the entire equation by 2 gives:
\[ 2x + y = 4 \]
Now we have the system:
- \(2x + y = 6\)
- \(2x + y = 4\)
Now, let's analyze these equations.
The first equation \(2x + y = 6\) can be rewritten to express \(y\) in terms of \(x\):
\[ y = 6 - 2x \]
The second equation \(2x + y = 4\) can be rewritten similarly:
\[ y = 4 - 2x \]
Now, we can see that both expressions for \(y\):
- \(y = 6 - 2x\)
- \(y = 4 - 2x\)
are equal for different values of \(y\). This indicates that the two lines represented by the equations are parallel and will never intersect.
Since the lines are parallel, there are no points that satisfy both equations. Therefore, the system has no solution.
The answer is:
0