Question

The image shows the graph of the system of linear equations {y=2x+2y=2x−4
{
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,−4)
(
0
,
−
4
)
open paren 0 comma negative 4 close paren

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

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

Both of these points are solutions.

Answer 1

To determine which points, if any, are solutions to the system of equations:

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

We can substitute the points to see if they satisfy both equations.

**Check the point \( (0, -4) \):**

Substituting \( x = 0 \) and \( y = -4 \) into both equations:

1. **First equation**:
\( y = 2(0) + 2 \)
\( -4 = 2 \) → **Not true**.

2. **Second equation**:
\( y = 2(0) - 4 \)
\( -4 = -4 \) → **True**.

Since this point only satisfies the second equation and not the first, \( (0, -4) \) is **not** a solution for the system.

---

**Check the point \( (0, 2) \):**

Substituting \( x = 0 \) and \( y = 2 \) into both equations:

1. **First equation**:
\( y = 2(0) + 2 \)
\( 2 = 2 \) → **True**.

2. **Second equation**:
\( y = 2(0) - 4 \)
\( 2 = -4 \) → **Not true**.

Since this point only satisfies the first equation and not the second, \( (0, 2) \) is **not** a solution for the system.

---

Since neither point is a solution, the correct response is:

**Neither of these points are solutions.**