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

Both of these points are solutions.
Both of these points are solutions.

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

Neither of these points are solutions.

Answer 1

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

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

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

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

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

Summary:

• The point \( (0, -4) \) is a solution to only the second equation.
• The point \( (0, 2) \) is a solution to only the first equation.

Therefore, the correct response is: Neither of these points are solutions.