Question

Solve the system of equations using whatever method you prefer.

y=x2+5x +9

y=x + 21

There are two correct answers.

(2 points)
Responses

(2,23)
(2,23)

(6, 27)
(6, 27)

(-6, 15)

Answer 1

To solve the system of equations given:

1. \( y = x^2 + 5x + 9 \)
2. \( y = x + 21 \)

We can set the two equations equal to each other since both are equal to \( y \):

\[
x^2 + 5x + 9 = x + 21
\]

Now, we will rearrange the equation to bring all terms to one side:

\[
x^2 + 5x + 9 - x - 21 = 0
\]

This simplifies to:

\[
x^2 + 4x - 12 = 0
\]

Next, we can factor this quadratic equation:

\[
(x + 6)(x - 2) = 0
\]

Setting each factor equal to zero gives us the solutions for \( x \):

1. \( x + 6 = 0 \) → \( x = -6 \)
2. \( x - 2 = 0 \) → \( x = 2 \)

Now we need to find the corresponding \( y \) values for both \( x \) values using either original equation. Using \( y = x + 21 \):

1. For \( x = -6 \):
\[
y = -6 + 21 = 15
\]
So, the first solution is \( (-6, 15) \).

2. For \( x = 2 \):
\[
y = 2 + 21 = 23
\]
So, the second solution is \( (2, 23) \).

Thus, the two solutions to the system of equations are:

- \( (-6, 15) \)
- \( (2, 23) \)

The correct answers from your provided responses are:

- \( (-6, 15) \)
- \( (2, 23) \)