Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x+8

Question

y
=
x
2
−
4
x
+
8
, y=3x−4
y
=
3
x
−
4
?(1 point)
Responses

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

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

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

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

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 can substitute the second equation into the first equation.

Setting the two equations equal to each other, we get:

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

Now, let's rearrange the equation:

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

This simplifies to:

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

Now, we can factor this quadratic equation:

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

Setting each factor to zero gives us the solutions:

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

Now we can find the corresponding \( y \) values by substituting these \( x \) values back into either original equation. We'll use \( y = 3x - 4 \):

For \( x = 3 \):

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

For \( x = 4 \):

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

Therefore, the solutions to the system of equations are \( (3, 5) \) and \( (4, 8) \).

The correct response is:

(4,8) and (3,5)