Question

Find the solution(s) to the system of equations.
Select all that apply.
Math
Description
1 y
5-|
y=x2-2x-3
y=x-3
0 (0, -3)
O (3, 0)
O (1, -4)
0 (0,3)

Answer 1

To find the solutions to the system of equations, we need to analyze the equations given. The two equations in this system are:

1. \( y = x^2 - 2x - 3 \)
2. \( y = x - 3 \)

We will solve for the intersection points (solutions) by setting the two equations equal to each other:

\[
x^2 - 2x - 3 = x - 3
\]

Next, rearranging this equation gives us:

\[
x^2 - 2x - 3 - x + 3 = 0
\]

Simplifying this yields:

\[
x^2 - 3x = 0
\]

Factoring out an \( x \):

\[
x(x - 3) = 0
\]

This gives us two solutions for \( x \):

1. \( x = 0 \)
2. \( x = 3 \)

Now, we can find the corresponding \( y \) values by substituting these \( x \) values into either equation. We'll use \( y = x - 3 \) for simplicity.

1. When \( x = 0 \):
\[
y = 0 - 3 = -3
\]
So, one solution is \( (0, -3) \).

2. When \( x = 3 \):
\[
y = 3 - 3 = 0
\]
So, the other solution is \( (3, 0) \).

Thus, the solutions to the system of equations are:

- \( (0, -3) \)
- \( (3, 0) \)

Now examining the provided options:
- \( (0, -3) \) ✅
- \( (3, 0) \) ✅
- \( (1, -4) \) ❌ (not a solution)
- \( (0, 3) \) ❌ (not a solution)

Therefore, the correct answers are:

- \( (0, -3) \)
- \( (3, 0) \)