Which of the following uses the substitution method to solve linear-quadratic system y=x2−4x+8, y=3x−4?(1 point)%0D%0AResponses%0D%0A%0D%0A(−4,−16) and (−3,−13)%0D%0Aleft parenthesis negative 4 comma negative 16 right parenthesis and left parenthesis negative 3 comma negative 13 right parenthesis%0D%0A%0D%0A(3,5)%0D%0Aleft parenthesis 3 comma 5 right parenthesis%0D%0A%0D%0A(4,8)%0D%0Aleft parenthesis 4 comma 8 right parenthesis%0D%0A%0D%0A(4,8) and (3,5)

Answer 1

To solve the linear-quadratic system using the substitution method, we have the following equations:

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

We will substitute the expression for \( y \) from the linear equation into the quadratic equation.

Substituting \( y = 3x - 4 \) into the quadratic equation:

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

Rearranging this equation gives:

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

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

Next, we factor the quadratic equation:

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

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

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

Now we substitute these values back into the linear equation to find the corresponding \( y \)-values.

For \( x = 3 \):

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

For \( x = 4 \):

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

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

From the provided responses, the correct answer is:

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