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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,−16)
and (−3,−13)
left parenthesis negative 4 comma negative 16 right parenthesis and left parenthesis negative 3 comma negative 13 right parenthesis

(4,8)

1 answer

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

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

Step 1: Substitute the second equation into the first

Since both equations are equal to \( y \), we can set them equal to each other:

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

Step 2: Rearrange the equation

To solve for \( x \), we will collect all terms on one side:

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

Combine like terms:

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

Step 3: Factor the quadratic equation

Now we need to factor the quadratic equation:

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

Step 4: Solve for \( x \)

This gives us two solutions for \( x \):

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

Step 5: Substitute \( x \) back to find \( y \)

Now we will substitute \( x \) back into the linear equation \( y = 3x - 4 \) to find the corresponding \( y \) values.

For \( x = 3 \):

\[ y = 3(3) - 4 = 9 - 4 = 5 \quad \Rightarrow \quad (3, 5) \]

For \( x = 4 \):

\[ y = 3(4) - 4 = 12 - 4 = 8 \quad \Rightarrow \quad (4, 8) \]

Final Solutions

The solutions to the system are:

  • \( (3, 5) \)
  • \( (4, 8) \)

Thus, the answer is:

(4, 8) and (3, 5)

So the response that corresponds to those solutions is:

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