To solve the system of equations using the substitution method, we have:
- \( y = x^2 - 4x + 8 \)
- \( y = 3x - 4 \)
We can substitute the second equation into the first equation.
Setting the two equations equal to each other, we get:
\[ 3x - 4 = x^2 - 4x + 8 \]
Now, let's rearrange the equation:
\[ 0 = x^2 - 4x - 3x + 8 + 4 \]
This simplifies to:
\[ 0 = x^2 - 7x + 12 \]
Now, we can factor this quadratic equation:
\[ 0 = (x - 3)(x - 4) \]
Setting each factor to zero gives us the solutions:
- \( x = 3 \)
- \( x = 4 \)
Now we can find the corresponding \( y \) values by substituting these \( x \) values back into either original equation. 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 \( (4, 8) \).
Therefore, the solutions to the system of equations are \( (3, 5) \) and \( (4, 8) \).
The correct response is:
(4,8) and (3,5)