Question

Which of the following uses the substitution method to solve linear-quadratic system y=x^2−4x+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 given linear-quadratic system using the substitution method, we have the following equations:

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

We will substitute the second equation into the first equation:

1. Set the equations equal:
\[
3x - 4 = x^2 - 4x + 8
\]

2. Rearrange the equation to bring all terms to one side:
\[
x^2 - 4x - 3x + 8 + 4 = 0
\]
\[
x^2 - 7x + 12 = 0
\]

3. Now, factor the quadratic:
\[
(x - 3)(x - 4) = 0
\]
This gives us:
\[
x = 3 \quad \text{or} \quad x = 4
\]

4. Next, substitute these \(x\) values back into one of the original equations to find the corresponding \(y\) values. We can use \(y = 3x - 4\):

- For \(x = 3\):
\[
y = 3(3) - 4 = 9 - 4 = 5 \quad \text{(Point: } (3, 5) \text{)}
\]

- For \(x = 4\):
\[
y = 3(4) - 4 = 12 - 4 = 8 \quad \text{(Point: } (4, 8) \text{)}
\]

Thus, the solution points for the system are \((3, 5)\) and \((4, 8)\).

Therefore, the correct response is:
\[
\text{(4, 8) and (3, 5)}
\]