The image shows the graph of the system of linear equations {y=2x+2y=2x−4

Question

{
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

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

Neither 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 will substitute the x-coordinates of each point into each equation and see if the corresponding y-values match.

1. For the point (0, 2):

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

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

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

   • Substitute \(x = 0\) into the first equation \(y = 2x + 2\): \[ y = 2(0) + 2 = 2 \] The point (0, -4) does not satisfy the first equation.

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

Conclusion:

• The point (0, 2) is a solution to the first equation but not to the second equation.
• The point (0, -4) is a solution to the second equation but not to the first equation.

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