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

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

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

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

(0,−4)

Answer 1

To determine whether the given points \((0, 2)\) and \((0, -4)\) are solutions to the system of equations \(y = 2x + 2\) and \(y = 2x - 4\), we will substitute the x-coordinates of the points into each equation and see if the corresponding y-values are satisfied.

1. For the point \((0, 2)\):

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

Since neither point satisfies both equations simultaneously, we conclude:

Neither of these points are solutions.