Use the image to answer the question.

Question

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

(1 point)
Responses

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

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

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

Both of these points are solutions.

Answer 1

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

1. For the point (0, 2):

   • Substitute \( x = 0 \) into the first equation: \[ y = 2(0) + 2 = 2 \] So the point (0, 2) satisfies the first equation.

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

   • Now substitute \( x = 0 \) into the second equation: \[ y = 2(0) - 4 = -4 \] So the point (0, -4) satisfies the second equation.

Since neither of the points (0, 2) nor (0, -4) satisfies both equations, the correct response is: Neither of these points are solutions.