Question

Solve the system by substitution. y=x2−2x+1 y=2x+1 (1 point) Responses (2,5) (3,7) (2,5) (3,7) (1,3) (-1,-1) (1,3) (-1,-1) (0,1) (4,9) (0,1) (4,9) (2,5) (1,3)

Answer 1

To solve the system of equations by substitution, we have:

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

We can set the two expressions for \( y \) equal to each other:

\[
x^2 - 2x + 1 = 2x + 1
\]

Now, simplify and solve for \( x \):

1. Subtract \( 2x + 1 \) from both sides:
\[
x^2 - 2x + 1 - 2x - 1 = 0
\]
\[
x^2 - 4x = 0
\]
2. Factor out an \( x \):
\[
x(x - 4) = 0
\]

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

\[
x = 0 \quad \text{or} \quad x = 4
\]

Now we will substitute these \( x \) values back into either equation to find the corresponding \( y \) values.

**For \( x = 0 \)**:
\[
y = 2(0) + 1 = 1
\]
So, the first solution is \( (0, 1) \).

**For \( x = 4 \)**:
\[
y = 2(4) + 1 = 9
\]
So, the second solution is \( (4, 9) \).

Thus, the solutions to the system of equations are:

\[
(0, 1) \quad \text{and} \quad (4, 9)
\]

Checking the response options, we have:

- (0, 1)
- (4, 9)

Both pairs match the calculations we completed. Thus, the final answer is:

\((0, 1)\) and \((4, 9)\) are both valid solutions.