Question

Which of the following uses the substitution method to solve linear-
quadratic system y = x^2 - 4x +8, y =3X -4? (1 point)
• (4,8) and (3, 5)
O (4,8)
О (3,5)
• (-4, -16) and (-3, -13)

Answer 1

To use the substitution method to solve the system of 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. Rearrange the equation:
\[
x^2 - 4x - 3x + 8 + 4 = 0
\]
\[
x^2 - 7x + 12 = 0
\]

3. To solve this quadratic equation, we can factor it:
\[
(x - 3)(x - 4) = 0
\]
So, \( x = 3 \) or \( x = 4 \).

4. Now, we substitute these \( x \) values back into one of the original equations to find the corresponding \( y \) values. We'll use \( y = 3x - 4 \).

- 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 the other solution is \( (4, 8) \).

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

Among the choices given, the correct answer using the substitution method to solve this linear-quadratic system is:

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