Question

Solve the system by substitution.

y=x^2−2x+1
y=2x+1
(1 point)
Responses

(2,5),(3,7)
open paren 2 comma 5 close paren comma open paren 3 comma 7 close paren

(1,3),(−1,−1)
open paren 1 comma 3 close paren comma open paren negative 1 comma negative 1 close paren

(0,1),(4,9)
open paren 0 comma 1 close paren comma open paren 4 comma 9 close paren

(2,5),(1,3)

Answer 1

To solve the system of equations using substitution, we need to replace one of the variables in one of the equations with an expression from the other equation.

Given the equations:
1. \( y = x^2 - 2x + 1 \) (Equation 1)
2. \( y = 2x + 1 \) (Equation 2)

We can substitute Equation 2 into Equation 1:

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

Now, let's simplify and solve for \( x \):

1. Rearrange the equation:
\[
x^2 - 2x - 2x + 1 - 1 = 0
\]
\[
x^2 - 4x = 0
\]

2. Factor the equation:
\[
x(x - 4) = 0
\]

This gives us two solutions:
\[
x = 0 \quad \text{or} \quad x = 4
\]

Now we will substitute these values back into Equation 2 to find the corresponding \( y \) values.

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

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

The solutions to the system of equations are:
\[
(0, 1) \quad \text{and} \quad (4, 9)
\]

Thus, the answer is:
\[
\text{(0,1),(4,9)}
\]