Question

Use the image to answer the 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 four points.

The image shows the graph of the following system of linear equations: {y=23x+5y=−x
. Which of the following points, if any, is a solution to the system?

(1 point)
Responses

(0,5)
open paren 0 comma 5 close paren

(2,3)
open paren 2 comma 3 close paren

(1,−1)
open paren 1 comma negative 1 close paren

(−3,3)

Answer 1

To determine which of the points is a solution to the system of equations given by \( y = 2x + 5 \) and \( y = -x \), we will substitute each point into both equations.

1. **Point (0, 5)**:
- For \( y = 2x + 5 \):
\[
5 = 2(0) + 5 \implies 5 = 5 \quad \text{(True)}
\]
- For \( y = -x \):
\[
5 = -(0) \implies 5 = 0 \quad \text{(False)}
\]
This point is **not a solution** to the system.

2. **Point (2, 3)**:
- For \( y = 2x + 5 \):
\[
3 = 2(2) + 5 \implies 3 = 4 + 5 \implies 3 = 9 \quad \text{(False)}
\]
This point is **not a solution** to the system.

3. **Point (1, -1)**:
- For \( y = 2x + 5 \):
\[
-1 = 2(1) + 5 \implies -1 = 2 + 5 \implies -1 = 7 \quad \text{(False)}
\]
This point is **not a solution** to the system.

4. **Point (-3, 3)**:
- For \( y = 2x + 5 \):
\[
3 = 2(-3) + 5 \implies 3 = -6 + 5 \implies 3 = -1 \quad \text{(False)}
\]
This point is **not a solution** to the system.

None of the provided points (0,5), (2,3), (1,-1), or (-3,3) satisfy both equations. Therefore, **none of the points is a solution to the system.**