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

2x+y=6

4x+2y=8

Line one Y=-2x+6
Line 2 y=-2x+4

Zero
Infinite
Two
One

1 answer

To analyze the system of equations:

  1. The first equation is \(2x + y = 6\). We can rearrange this into slope-intercept form (y = mx + b): \[ y = -2x + 6 \]

  2. The second equation is \(4x + 2y = 8\). We can also rearrange this into slope-intercept form: \[ 2y = -4x + 8 \quad \Rightarrow \quad y = -2x + 4 \]

Now we have the two equations in slope-intercept form:

  • Line 1: \(y = -2x + 6\)
  • Line 2: \(y = -2x + 4\)

Both lines have the same slope (-2), which indicates they are parallel. Because the y-intercepts of the two lines are different (6 for the first line and 4 for the second line), the lines will never intersect.

Since the lines are parallel and do not coincide, there are no points of intersection. Therefore, there are no solutions to this system of equations.

The correct answer is: Zero.