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

Question

2x+y=6

4x+2y=8
(1 point)
Responses

2
2

1
1

infinite
infinite

0

Answer 1

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

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

First, we can simplify Equation 2:

Divide the entire Equation 2 by 2:

\[ \frac{4x + 2y}{2} = \frac{8}{2} \]

This gives us:

\[ 2x + y = 4 \]

Now, we have the system:

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

Next, notice that the left sides of both equations are identical, but their right sides are different (6 and 4). This means that the two equations represent parallel lines in the Cartesian plane, which do not intersect.

Since the lines are parallel, there are no points \((x, y)\) that satisfy both equations simultaneously. Therefore, the system has no solutions.

The answer is:

0