Question

Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x+8 , y=3x−4 ?(1 point)
Responses

(−4,−16) and (−3,−13)
left parenthesis negative 4 comma negative 16 right parenthesis and left parenthesis negative 3 comma negative 13 right parenthesis

(4,8)
left parenthesis 4 comma 8 right parenthesis

(4,8) and (3,5)
left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis

(3,5)

Answer 1

To solve the linear-quadratic system using the substitution method, we have the following equations:

1. \( y = x^2 - 4x + 8 \) (quadratic)
2. \( y = 3x - 4 \) (linear)

We can substitute the expression for \( y \) from the second equation into the first equation:

1. Substitute \( 3x - 4 \) for \( y \) in the first equation:

\[
3x - 4 = x^2 - 4x + 8
\]

2. Rearranging gives us:

\[
x^2 - 4x - 3x + 8 + 4 = 0
\]

This simplifies to:

\[
x^2 - 7x + 12 = 0
\]

3. Now, we can factor or use the quadratic formula to solve for \( x \):

\[
(x - 3)(x - 4) = 0
\]

Thus, the solutions are:

\[
x = 3 \quad \text{and} \quad x = 4
\]

4. We now find the corresponding \( y \) values by substituting back into \( y = 3x - 4 \):

- For \( x = 3 \):

\[
y = 3(3) - 4 = 9 - 4 = 5 \quad \Rightarrow \quad (3, 5)
\]

- For \( x = 4 \):

\[
y = 3(4) - 4 = 12 - 4 = 8 \quad \Rightarrow \quad (4, 8)
\]

The solutions to the system are:

\[
(3, 5) \quad \text{and} \quad (4, 8)
\]

Therefore, the correct response is:

**(4,8) and (3,5)**.