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The image shows the graph of the system of linear equations {y = 2x + 2. Which of the following points, {y = 2x - 4. if any, are solutions to the system of equations?

A: Neither of these points are solutions

B: Both of these points are solutions.

C: ( 0,2 )

D: ( 0, -4)

1 answer

To determine whether the points \( (0, 2) \) and \( (0, -4) \) are solutions to the system of equations \( y = 2x + 2 \) and \( y = 2x - 4 \), we can evaluate each point in both equations.

  1. For the point \( (0, 2) \):

    • Substitute \( x = 0 \) into the first equation: \[ y = 2(0) + 2 = 2 \]
      • So, \( (0, 2) \) satisfies the first equation.
    • Now, substitute \( x = 0 \) into the second equation: \[ y = 2(0) - 4 = -4 \]
      • So, \( (0, 2) \) does not satisfy the second equation.

  2. For the point \( (0, -4) \):

    • Substitute \( x = 0 \) into the first equation: \[ y = 2(0) + 2 = 2 \]
      • So, \( (0, -4) \) does not satisfy the first equation.
    • Now, substitute \( x = 0 \) into the second equation: \[ y = 2(0) - 4 = -4 \]
      • So, \( (0, -4) \) satisfies the second equation.

In summary:

  • \( (0, 2) \): satisfies \( y = 2x + 2 \) but not \( y = 2x - 4 \).
  • \( (0, -4) \): satisfies \( y = 2x - 4 \) but not \( y = 2x + 2 \).

Neither of the points are solutions to the entire system of equations. Therefore, the answer is:

A: Neither of these points are solutions.