Question

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

(4,8)

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

left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis

Answer 1

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

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

We'll start by substituting the expression for \( y \) from the second equation into the first equation:

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

Next, we rearrange this equation to form a standard quadratic equation:

\[
0 = x^2 - 4x - 3x + 8 + 4
\]
\[
0 = x^2 - 7x + 12
\]

Now, we can factor the quadratic:

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

Setting each factor to zero gives us:

\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]

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

For \( x = 3 \):
\[
y = 3(3) - 4 = 9 - 4 = 5
\]

For \( x = 4 \):
\[
y = 3(4) - 4 = 12 - 4 = 8
\]

So the solutions to the system of equations are:
- For \( x = 3 \), \( y = 5 \): \( (3, 5) \)
- For \( x = 4 \), \( y = 8 \): \( (4, 8) \)

In conclusion, the solutions that use the substitution method to solve the system are:

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

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