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

(3,5)
left parenthesis 3 comma 5 right parenthesis

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

(4,8)

Answer 1

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

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

We can substitute the second equation into the first equation. This gives us:

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

Next, we rearrange the equation:

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

This simplifies to:

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

Now, we factor the quadratic equation:

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

Thus, we have two solutions for \( x \):

1. \( x = 3 \)
2. \( x = 4 \)

Now, we can find the corresponding \( y \) values using either of the original equations (we'll use the linear equation \( y = 3x - 4 \)):

1. For \( x = 3 \):
\[
y = 3(3) - 4 = 9 - 4 = 5
\]
So one solution is \( (3, 5) \).

2. For \( x = 4 \):
\[
y = 3(4) - 4 = 12 - 4 = 8
\]
So the other solution is \( (4, 8) \).

Thus, the solutions to the system are \( (4, 8) \) and \( (3, 5) \).

The correct response is:
**(4,8) and (3,5)**.