Question

Solving Simultaneous Equations Using Substitution Quick Check
4 of 54 of 5 Items
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) and (3,5)
left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis

(4,8)
left parenthesis 4 comma 8 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

(3,5)
left parenthesis 3 comma 5 right parenthesis
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Answer 1

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

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

We can set the two equations equal to each other since both equal \( y \):

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

Now, let's rearrange it to form a standard quadratic equation:

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

Now, factor the quadratic:

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

Setting each factor to zero gives us the values for \( x \):

1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x - 4 = 0 \) → \( x = 4 \)

Next, we can find the corresponding \( y \) values by substituting \( x \) back into either of the original equations. We'll use the second equation for simplicity:

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 second solution is \( (4, 8) \).

Thus, the solutions to the system are:

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

The correct response is:

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

or in the requested format:

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