Use the image to answer the question.

Question

A coordinate graph goes from negative 10 to 10 in 1-unit increments on both the x- and y-axes. Two linear equations are graphed, along with two points.

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

(1 point)
Responses

(0,−4)
open paren 0 comma negative 4 close paren

Neither of these points are solutions.
Neither of these points are solutions.

(0,2)
open paren 0 comma 2 close paren

Both of these points are solutions.

Answer 1

To determine if the points \( (0, -4) \) and \( (0, 2) \) are solutions to the system of equations given by \( y = 2x + 2 \) and \( y = 2x - 4 \), we can substitute the x-coordinate of each point into the equations and see if the corresponding y-coordinate matches.

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

   • Substitute \( x = 0 \) into the first equation: \[ y = 2(0) + 2 = 2 \] This does not match \( -4 \).
   • Substitute \( x = 0 \) into the second equation: \[ y = 2(0) - 4 = -4 \] This matches \( -4 \).
   • Therefore, \( (0, -4) \) is a solution to the second equation but not to the first.

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

   • Substitute \( x = 0 \) into the first equation: \[ y = 2(0) + 2 = 2 \] This matches \( 2 \).
   • Substitute \( x = 0 \) into the second equation: \[ y = 2(0) - 4 = -4 \] This does not match \( 2 \).
   • Therefore, \( (0, 2) \) is a solution to the first equation but not to the second.

Since neither point is a solution to both equations of the system, the correct response is:

Neither of these points are solutions.