Question

Use the substitution method to find the solutions to the linea-quadratic system

y = 2x − 3


y = x2 + 6x − 3




(1 point)
Responses

(1, -1) and (2, 1)
(1, -1) and (2, 1)

(-4, -11) and (0, -3)
(-4, -11) and (0, -3)

(-4, -11)
(-4, -11)

(0, -3)

Answer 1

To solve the system of equations using the substitution method, we start with the given equations:

1. \( y = 2x - 3 \)
2. \( y = x^2 + 6x - 3 \)

Since we have \( y \) expressed in terms of \( x \) in the first equation, we can substitute \( 2x - 3 \) for \( y \) in the second equation.

Substituting, we have:

\[
2x - 3 = x^2 + 6x - 3
\]

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

1. Add \( 3 \) to both sides:

\[
2x = x^2 + 6x
\]

2. Rearranging gives us:

\[
0 = x^2 + 6x - 2x
\]
\[
0 = x^2 + 4x
\]

3. Factor the quadratic equation:

\[
0 = x(x + 4)
\]

Setting each factor to zero gives:

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

Now, we can find the corresponding \( y \) values by substituting back into the first equation:

1. For \( x = 0 \):
\[
y = 2(0) - 3 = -3
\]
So, one solution is \( (0, -3) \).

2. For \( x = -4 \):
\[
y = 2(-4) - 3 = -8 - 3 = -11
\]
So, the other solution is \( (-4, -11) \).

Thus, the solutions to the system are:

- \( (0, -3) \)
- \( (-4, -11) \)

Considering the possible responses given:

- (1, -1) and (2, 1)
- (-4, -11) and (0, -3)
- (-4, -11)
- (0, -3)

The correct choice is:

**(-4, -11) and (0, -3)**.