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

(4,8)
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

Answer 1

To solve the system of equations using the substitution method, we can substitute the expression for \(y\) from the second equation into the first equation.

We have the equations:

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

Substituting \(3x - 4\) for \(y\) in the first equation:

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

Now, rearranging the equation:

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

Next, we can factor the quadratic:

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

Setting each factor to zero gives:

\[
x - 3 = 0 \quad \text{or} \quad x - 4 = 0
\]

Thus, \(x = 3\) or \(x = 4\).

Now we can find the corresponding \(y\) values for each \(x\):

1. For \(x = 3\):
\[
y = 3(3) - 4 = 9 - 4 = 5 \quad \Rightarrow \quad (3, 5)
\]

2. For \(x = 4\):
\[
y = 3(4) - 4 = 12 - 4 = 8 \quad \Rightarrow \quad (4, 8)
\]

So the solutions to the system are \( (3, 5) \) and \( (4, 8) \).

Therefore, the correct response is:

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