To solve the system of equations using the substitution method, we have:
- \( y = x^2 - 4x + 8 \)
- \( y = 3x - 4 \)
We'll start by substituting the expression for \( y \) from the second equation into the first equation:
\[ 3x - 4 = x^2 - 4x + 8 \]
Next, we rearrange this equation to form a standard quadratic equation:
\[ 0 = x^2 - 4x - 3x + 8 + 4 \] \[ 0 = x^2 - 7x + 12 \]
Now, we can factor the quadratic:
\[ 0 = (x - 3)(x - 4) \]
Setting each factor to zero gives us:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
Now we substitute these \( x \) values back into the equation \( y = 3x - 4 \) to find the corresponding \( y \) values.
For \( x = 3 \): \[ y = 3(3) - 4 = 9 - 4 = 5 \]
For \( x = 4 \): \[ y = 3(4) - 4 = 12 - 4 = 8 \]
So the solutions to the system of equations are:
- For \( x = 3 \), \( y = 5 \): \( (3, 5) \)
- For \( x = 4 \), \( y = 8 \): \( (4, 8) \)
In conclusion, the solutions that use the substitution method to solve the system are:
\[ (4, 8) \text{ and } (3, 5) \]
Thus, the correct selection is: \[ (4, 8) \text{ and } (3, 5) \]