Question

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

(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,−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)

Answer 1

To solve the system of equations using the substitution method, we have the equations:

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

### Step 1: Substitute the second equation into the first

Since both equations are equal to \( y \), we can set them equal to each other:

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

### Step 2: Rearrange the equation

To solve for \( x \), we will collect all terms on one side:

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

Combine like terms:

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

### Step 3: Factor the quadratic equation

Now we need to factor the quadratic equation:

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

### Step 4: Solve for \( x \)

This gives us two solutions for \( x \):

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

### Step 5: Substitute \( x \) back to find \( y \)

Now we will substitute \( x \) back into the linear equation \( y = 3x - 4 \) to find the corresponding \( y \) values.

For \( x = 3 \):

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

For \( x = 4 \):

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

### Final Solutions

The solutions to the system are:

- \( (3, 5) \)
- \( (4, 8) \)

Thus, the answer is:

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

So the response that corresponds to those solutions is:

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