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

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

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

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

(0,−4)

Answer 1

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 substitute the x-values of each point into both equations and see if the corresponding y-values match.

1. For the point (0, 2):

   • Substituting \( x = 0 \) in the first equation \( y = 2x + 2 \): \[ y = 2(0) + 2 = 0 + 2 = 2 \] This point lies on the first equation.

   • Substituting \( x = 0 \) in the second equation \( y = 2x - 4 \): \[ y = 2(0) - 4 = 0 - 4 = -4 \] This point does not lie on the second equation.

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

   • Substituting \( x = 0 \) in the first equation \( y = 2x + 2 \): \[ y = 2(0) + 2 = 0 + 2 = 2 \] This point does not lie on the first equation.

   • Substituting \( x = 0 \) in the second equation \( y = 2x - 4 \): \[ y = 2(0) - 4 = 0 - 4 = -4 \] This point lies on the second equation.

Since (0, 2) satisfies the first equation but not the second, and (0, -4) satisfies the second equation but not the first, neither point is a solution to the system of equations.

Therefore, the correct response is:

Neither of these points are solutions.