Which of the following uses the substitution method to solve linear-quadratic system y=x^2−4x+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) and (3,5) left parenthesis 4 comma 8 right parenthesis and left parenthesis 3 comma 5 right parenthesis (4,8)

1 answer

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

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

We will substitute the second equation into the first equation:

  1. Set the equations equal: \[ 3x - 4 = x^2 - 4x + 8 \]

  2. Rearrange the equation to bring all terms to one side: \[ x^2 - 4x - 3x + 8 + 4 = 0 \] \[ x^2 - 7x + 12 = 0 \]

  3. Now, factor the quadratic: \[ (x - 3)(x - 4) = 0 \] This gives us: \[ x = 3 \quad \text{or} \quad x = 4 \]

  4. Next, substitute these \(x\) values back into one of the original equations to find the corresponding \(y\) values. We can use \(y = 3x - 4\):

    • For \(x = 3\): \[ y = 3(3) - 4 = 9 - 4 = 5 \quad \text{(Point: } (3, 5) \text{)} \]

    • For \(x = 4\): \[ y = 3(4) - 4 = 12 - 4 = 8 \quad \text{(Point: } (4, 8) \text{)} \]

Thus, the solution points for the system are \((3, 5)\) and \((4, 8)\).

Therefore, the correct response is: \[ \text{(4, 8) and (3, 5)} \]