Determine the number of solutions to this system of linear equations:

To determine the number of solutions to the system of equations:

1. \( 2x + y = 6 \)
2. \( 4x + 2y = 8 \)

First, let's take a closer look at the second equation. We can rewrite it to see if it simplifies correctly.

Starting with the second equation:

\[ 4x + 2y = 8 \]

We can divide the entire equation by 2:

\[ 2x + y = 4 \]

Now, we can rewrite our system of equations:

1. \( 2x + y = 6 \)
2. \( 2x + y = 4 \)

Now we see that the left-hand sides of both equations are the same, but the right-hand sides differ (6 and 4). This means that the two equations represent parallel lines that will never intersect.

Since the two equations represent parallel lines, there are no solutions to the system of equations.

Thus, the number of solutions is:

0

Yes, let’s confirm the findings for the system of equations. We have:

1. \( 2x + y = 6 \)
2. \( 4x + 2y = 8 \)

We simplified the second equation properly:

\[ 4x + 2y = 8 \implies 2x + y = 4 \quad \text{(by dividing everything by 2)} \]

Now we can write the system as:

1. \( 2x + y = 6 \)
2. \( 2x + y = 4 \)

Now, let's analyze these two equations:

1. Equation 1: \( 2x + y = 6 \) can be rewritten as \( y = 6 - 2x \).
2. Equation 2: \( 2x + y = 4 \) can be rewritten as \( y = 4 - 2x \).

The left sides of the equations are the same, indicating they represent lines with the same slope but different y-intercepts (6 and 4). Consequently, they are parallel lines that do not intersect.

Since they do not intersect, the system has no solutions. Therefore, the conclusion remains that the number of solutions to the system of equations is:

0