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

Answer 1

To solve the system of equations using the substitution method for the equations \( y = x^2 - 4x + 8 \) and \( y = 3x - 4 \), we can set the two equations equal to each other since they both equal \( y \).

1. Set the equations equal:

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

2. Rearranging the equation gives:

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

3. Next, we can factor the quadratic:

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

4. Setting each factor to zero gives us the values of \( x \):

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

5. Now we substitute back to find \( y \) for both values of \( x \):

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

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

The correct response is:
\((4,8)\) and \((3,5)\).