Determine the number of solutions to this system of linear equations

Question

2x+y=6

4x+2y=8


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(1 point)
Responses

0
0

1
1

2
2

infinite

Answer 1

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

1. The first equation is \(2x + y = 6\).
2. The second equation is \(4x + 2y = 8\).

Let's simplify the second equation. We can divide all terms by 2:

\[ 4x + 2y = 8 \implies 2x + y = 4 \]

Now we have the following system:

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

Next, let's analyze the two equations:

• The first equation states that \(y = 6 - 2x\).
• The second equation states that \(y = 4 - 2x\).

Since both equations have the same left-hand side (\(2x + y\)), but different right-hand sides (6 and 4), they represent two parallel lines in the coordinate plane. Parallel lines never intersect, meaning there are no points \((x, y)\) that satisfy both equations simultaneously.

Thus, the number of solutions to this system of equations is:

0.