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.

Question

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

(0,2)

Answer 1

To determine if the points (0, -4) and (0, 2) are solutions to the given system of linear equations, we will substitute the x-coordinates of the points into both equations and see if the resulting y-values match.

The equations are:

1. \(y = 2x + 2\)
2. \(y = 2x - 4\)

Now, substitute \(x = 0\) into both equations:

For the point (0, -4):

1. First Equation:
   \(y = 2(0) + 2 = 2\)
   So, the first equation gives \(y = 2\), not -4.

2. Second Equation:
   \(y = 2(0) - 4 = -4\)
   So, the second equation gives \(y = -4\).

The point (0, -4) satisfies the second equation but not the first equation. Thus, (0, -4) is not a solution to the system.

For the point (0, 2):

1. First Equation:
   \(y = 2(0) + 2 = 2\)
   So, the first equation gives \(y = 2\), which matches.

2. Second Equation:
   \(y = 2(0) - 4 = -4\)
   So, the second equation gives \(y = -4\), which does not match.

The point (0, 2) satisfies the first equation but not the second equation. Thus, (0, 2) is not a solution to the system.

Conclusion:

Since neither (0, -4) nor (0, 2) are solutions to both equations, the correct response is:

Neither of these points are solutions.