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

2x+y=6

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

1
1

0
0

2
2

infinite
infinite

1 answer

To determine the number of solutions to the given system of linear equations, we can analyze the equations:

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

Let's manipulate the second equation to see if it's dependent on the first one. We can divide the entire second equation by 2:

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

This simplifies to:

\[ 2x + y = 4 \]

Now we can clearly see the system of equations:

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

We can see that these two equations are inconsistent, as their left-hand sides are identical while the right-hand sides are different. This means that there is no point \((x, y)\) that satisfies both equations simultaneously.

Therefore, the number of solutions to the system of equations is:

0 (no solutions).