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)

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

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

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

(0,2)

Answer 1

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

1. Point (0, -4):

   • For the first equation \( y = 2x + 2 \):
     • Substitute \( x = 0 \): \[ y = 2(0) + 2 = 2 \]
     • The y-value is 2, which does not match -4.
   • For the second equation \( y = 2x - 4 \):
     • Substitute \( x = 0 \): \[ y = 2(0) - 4 = -4 \]
     • The y-value is -4, which matches. Thus, (0, -4) is a solution to the second equation but not to the first.

2. Point (0, 2):

   • For the first equation \( y = 2x + 2 \):
     • Substitute \( x = 0 \): \[ y = 2(0) + 2 = 2 \]
     • The y-value is 2, which matches 2.
   • For the second equation \( y = 2x - 4 \):
     • Substitute \( x = 0 \): \[ y = 2(0) - 4 = -4 \]
     • The y-value is -4, which does not match 2. Thus, (0, 2) is a solution to the first equation but not to the second.

Since neither of the points (0, -4) nor (0, 2) satisfies both equations simultaneously, the answer is:

Neither of these points are solutions.